
Class QCS_1 



Book. M$l 

Copyright N° 



COPYRIGHT DEPOSIT, 



A TEXT-BOOK ' 




ON 



STATIC ELECTRICITY 



HOB ART MASON, B. S., E. E. 

Lat 3 Assistant in Electrical Engineering at the Polytechnic Institute of Brook- 
lyn and Associate of the American Institute of Electrical Engineers. 



NEW YORK 

McGRAW PUBLISHING COMPANY 

114 Liberty Street 

1904 



4--*> 



LIBRARY of CONGRESS 

Two Copies Received 

MAY 9 1904 

Copyright Entry 

Y/Ksovv)- L> - l q f 
CLASS^ 0-XXc. No. 

COPY B 



Copyrighted, 1904, 

BY THE 

McGRAW PUBLISEINQ COMPANY, 
New York. 



PREFACE 






The author was led to the preparation of this volume by 
the apparent lack of any suitable work of similar descrip- 
tion. The subject of Static Electricity is touched on in the 
average " Physics " or " Natural Philosophy " in a most 
gingerly fashion ; text-books devoted entirely to electricity 
seem to be either " Electric Machines " or "Alternating 
Currents/' and avoid the subject entirely save for some 
slight consideration of the electrodynamic aspects of the 
condenser ; some of the very numerous " Electricity and 
Magnetisms " have a certain amount of matter, not happily 
arranged for either a text-book or a book of reference, and 
are often unfortunately loose in statement and definition ; 
and it would seem that the only adequate literature on the 
subject lay in papers so profound as to be practically in- 
accessible to the ordinary student or to him who desires to 
make a reference without reading whole volumes. 

And so the author, who has had perforce to do a certain 
amount of reading on the subject, ventures to put forth this 
work. There is little or nothing original in it, but it js 
hoped that by the judicious combining of the wisdom of 
others in lucid and orderly fashion, and by keeping strictly 
to the subject of Electrostatics, he has made a volume val- 
uable to the student, the physicist, and the casual seeker 
after electrostatic truths. 

There has been a constant endeavor throughout the work 
to make the reading as simple as is consistent with exact- 



iv PREFACE. 

ness and a reasonable terseness. The methods of the cal- 
culus have been avoided save where demanded for pre- 
ciseness or compactness of expression. The copious index 
will enhance its value as a book of reference. 

In the Appendix have been put two tables and a few 
paragraphs which, while not strictly appertaining to the 
subject of this book, will, it is believed, be of interest to 
him who is interested in electrostatics. Many students of 
electricity are hazy regarding, if not quite ignorant of these 
two subjects, the physical dimensions of units, and the rela- 
tion between the units of the electrostatic series and those 
of the electromagnetic system. 

Acknowledgment of indebtedness is herewith made to 
those manufacturing concerns that have given information 
regarding, and loaned electrotypes of their apparatus. 

Hobaet Mason. 
Brooklyn, "N, Y., November, 1903. 



CONTENTS. 



CHAPTER I. PAGE . 

General Phenomena 1 

1. Historical 1 

2. Attraction 1 

3. Electrification and Charge 3 

4. Repulsion 4 

5. Positive and Negative Charges 5 

6. Electrostatic Series 7 

7. Volta's Series 8 

8. Conductors and Insulators 8 

9. Influence 12 

10. Bound and Free Charges 14 

11. Electroscopes 17 

CHAPTER II. 

The Electrostatic Field 20 

12. Coulomb's Law 20 

13. Quantity 21 

14. Surface Density 22 

15. The Electrostatic Field 26 

16. Intensity 27 

17. Potential 28 

18. Flux through a Closed Surface 33 

19. No Charge within a Conductor 35 

20. Charges in a Cavity 37 

21. Tubes of Force 39 

22. Intensity Outside a Charged Surface 41 

23. Stress on Electrified Surface 42 

24. Attraction Due to Electrified Plane 44 

CHAPTER III. 

Capacity 46 

25. Definition 46 

26. Capacity of a Sphere 46 

27. Principle of the Condenser 47 

28. Leyden Jar 49 

29. Seat of the Charge 52 

30. Specific Inductive Capacity 52 

V 



CONTENTS. 



Capacity — Continued. page. 

31. Mechanical Stresses Due to Charge 54 

32. Dielectric Hysteresis 56 

33. Calculation of Capacity 58 

34. Energy of Charged Condenser 64 

35. Commercial Condensers 65 

36. Electrolytic Condensers 68 

37. Connection of Condensers 70 

CHAPTER IV. 

Experimental Measurement of Capacity 73 

38. Measurement of Capacities 73 

39. Comparison of Capacities 77 

CHAPTER V. 

Instruments Used in Electrostatics 84 

40. Ballistic Galvanometer 84 

41. Calibration of Ballistic Galvanometer 86 

42. Kelvin's Absolute Electrometer 93 

43. Quadrant Electrometers 96 

44. Capillary Electrometers 104 

45. Electrostatic Voltmeters 109 

46. Electrostatic Ground Detectors 112 

CHAPTER VI. 

High Potential Static Generators 117 

47. Friction Machines 117 

48. The Toepler Machine 120 

49. The Holtz Machine 126 

50. The Wimshurst Machine 132 

51. Static Machines as Motors 134 

52. The Dropping Generator 135 

53. Discharges of the Static Machine 137 

APPENDIX. 

A. Dimensions of Units 143 

B. Ratio of the Units 148 



STATIC ELEOTEICITT. 



CHAPTEK I. 

GENERAL PHENOMENA. 

1. Historical. — Electricity is the name given to that 
which produces electrical phenomena, but while electrical 
phenomena are well known and understood, the nature 
of electricity is as yet unknown. The term " electricity " 
comes from the Greek rjXsKrpov^ amber, since, as Thales 
of Miletus (600 B. C.) informs us, the ancient Greeks 
recognized the fact that amber when rubbed attracted 
light particles to it, or as we would say, became electrified. 
Until about 1G00 amber and jet were the only substances 
that were known to have this property. Then Dr. Gil- 
bert discovered that many substances, such as glass, resin, 
sulphur, and others, also possessed this property, so the 
seventeenth century marks the real beginning of electrical 
science. 

2. Attraction. — The most obvious property of an 
electrified body is that of attraction. This property is 
easily illustrated by rubbing a piece of amber, as did the 
Greeks, and noting that it will then attract dust, chaff, 
small bits of straw and paper, or a pith ball suspended 
on a thread. In experiments of this nature it is usual 
to use not amber for the body to be electrified, but a 
stick of vulcanite (hard rubber) which is rubbed by a 

1 



2 STATIC ELECTRICITY. 

piece of cat's fur, or a glass rod nibbed by a silk cloth. 
The property of attraction can further be well shown by 
holding the electrified stick of vulcanite near a thin 
stream of water, as in Fig. 1, and noticing that stream 
i3 deflected so as to flow nearer the vulcanite. Since the 
presence of moisture precludes the possibility of success- 
ful electrification, care must be taken in this experiment 
that neither the stream nor any spray from it be allowed 
to touch the electrified bodv. 




Fig. 1. 

Electrostatic attraction can also be shown by briskly 
drawing a sheet of paper between the coat-sleeve and the 
side. The paper is then electrified, and if laid against 
the wall without more handling than necessary, it will 
so strongly attract the wall as to sustain its own weight 
until its electrification is exhausted, perhaps for a quarter 
of an hour. Like all other electrostatic experiments, this 
one will work more satisfactorily on a dry day, and even 
a reasonable amount of humidity may cause it to fail 
entirely. 

Those who have dry, light hair have often observed 
that, on running the comb through it, the hair is attracted 



GENERAL PHENOMENA. 3 

by the comb even after the latter has been removed some 
inches from the head. This happens only on dry days 
and with combs of hard rubber or similar material. Metal 
combs cannot be made to produce this effect. It some- 
times occurs that on brushing the clothes it seems im- 
possible to remove the dust, and the more vigorously the 
brush is wielded the more tenaciously the dust seems to 
stick. This is due to the electrification of the cloth and 
the consequent attraction of the dust particles. 

3. Electrification and Charge. — When a body has ac- 
quired the power of attraction illustrated in the last sec- 
tion it is said to be electrified, or to carry a charge of 
electricity. In the consideration of this matter it is con- 
venient to think of the electricity as something material 
that is laid on to the body in charging it. The quantity 
laid on is a measure of the quantity of the charge, and 
the thickness with which it covers the body is a measure 
of the surface density, sometimes called tension, of the 
charge. It must be remembered that this idea is entirely 
fictitious, but it affords a convenient way of gaining a com- 
prehension of the actions and relations of electricity which 
in itself is uncomprehended. 

When a charged stick of vulcanite attracts to itself a 
piece of paper, energy must be expended in moving that 
paper. Since the vulcanite was powerless to move the 
paper before it was electrified it must have received the 
energy in the process of charging, since by the law of 
conservation of energy, energy cannot be created, but only 
transferred and transformed. If the vulcanite had been 
charged by rubbing with cat's fur, enough energy was 
expended in the rubbing to account for the energy the 
vulcanite afterward showed itself to possess when it at- 
tracted the paper chip. If the vulcanite had been electri- 



4 STATIC ELECTRICITY. 

fied or charged by contact with another charged body, it 
would have received its energy along with its charge from 
that body and the sum of energy in the two bodies would 
then be equal to the energy the original charged body 
had at first. Thus it is seen that when a body is electri- 
fied or charged it possesses an increased energy, and that 
energy must have been expended to bring it to the state 
of electrification. 

4. Repulsion. — Attraction is not the only property of 
a charged body. If a pith ball suspended on a thread be 
attracted by an electrified glass rod, and be allowed to 
touch that rod, it will immediately be repelled and held 
at a distance. The reason is that the pith ball on touch- 
ing the glass received some of the charge from the glass, 
and bodies similarly charged repel one another. 

This repulsion of similarly electrified bodies is shown 
by bringing a charged ebonite rod near a jet from which 
a small stream of water is issuing. Through the influence 
(§ 9) of the charged rod the water particles become them- 
selves charged each with a similar charge, and therefore 
repel each other and the stream breaks up into a spray. 
In a capillary tube, through which water will not ordi- 
narily flow, the fluid can sometimes be made to pass by 
the presence of an electrified body in the neighborhood. 
The particles are then similarly charged and have a ten- 
dency to repel each other. As a consequence the surface 
tension of the liquid is lessened and the water. will flow. 
Similarly a fine stream of water may be made to issue 
from a pin hole in the bottom of a charged containing 
vessel through which it will not pass when not electrified. 

All electrified bodies do not repel one another, but only 
those that are similarly electrified. To show this, let a 
vulcanite rod rubbed by a woolen cloth be approached to 



GENERAL PHENOMENA. 5 

a suspended pith ball. The latter will be attracted until 
it has touched the rod and received some of the charge, 
when it will be repelled. Now let a glass rod rubbed by 
a silk cloth be approached to the pith ball and it will be 
very strongly attracted. In fact the attraction may be 
noticed to be stronger than it would have been had the 
ball not been previously charged by contact with the vul- 
canite rod. Let, however, the ball touch the glass rod 




Fig. 2. 

and it gives up its previous charge and takes on some 
of that of the glass, and is, in consequence, repelled. 
The pith ball is now strongly attracted by the vulcanite 
rod. This experiment shows that while bodies similarly 
charged repel one another, two bodies charged by different 
means may attract each other. 

5. Positive and Negative Charges. — In order to investi- 
gate the different kinds of electrification, let a glass rod 
be swung in a stirrup by a thread, as in Fig. 2. Charge 



6 STATIC ELECTRICITY. 

the rod by rubbing briskly with a silk cloth. If now 
another glass rod, also charged by rubbing with a silk 
cloth, be approached to the suspended rod, a repulsion 
is observed. Let, however, a rod of vulcanite, electrified 
by rubbing with a woolen cloth, be approached, and an 
attraction is observed. Substituting in turn sulphur, met- 
als, resin, and celluloid, each rubbed by silk, and attraction 
is observed. Clearly then, there must be two classes of elec- 
trification, one attracting the charge on the suspended 
rod, the other repelling it. In former years, on the suppo- 
sition that glass could have but the one kind of electrifica- 
tion, the two kinds of electrification were called respec- 
tively vitreous and resinous. The unsuitability of these 
terms is shown by the following experiment. Let the glass 
rod in the stirrup (Fig. 2) be rubbed by a silk cloth and 
another glass rod be rubbed by a woolen cloth or a piece of 
fur. On approaching the second to the first an attraction 
is observed. From this we see that the kind of electrifica- 
tion resulting from the rubbing of a substance depends 
not only on that substance, but also on the material with 
which it is rubbed. To distinguish the two kinds of 
electrification, it has been arbitrarily decided that the 
charge on a glass rod resulting from rubbing it with a silk 
cloth shall be called a positive charge; that all charged 
bodies which this glass rod tends to repel are also posi- 
tively charged, and that those charged bodies which it 
tends to attract are negatively charged. 

It is further true that neither kind of charge can be 
produced alone. When a vulcanite rod is electrified by 
rubbing with a woolen cloth, the cloth receives a positive 
charge exactly corresponding to the negative charge on 
the rod. That the cloth is charged may be shown by 
some of the means employed in the preceding experiments, 



GENERAL PHENOMENA. 7 

or more easily by means of the electroscope, an apparatus 
described at the end of this chapter. 

6. Electrostatic Series. — 'By continued experimenting, 
rubbing different substances together in pairs, and noting 
which is positively and which negatively charged, the 
following condition is found to exist. If A be charged 
positively when rubbed by B, and if B be charged posi- 
tively when rubbed by C ', then A will be charged posi- 
tively when rubbed by C. This condition makes it 
possible to arrange a series of substances in such order 
that each, when rubbed by any one preceding it in the list, 
becomes negatively charged, and when rubbed by any one 
succeeding it in the list becomes positively charged. The 
following series is so arranged that if any two of the 
substances mentioned be rubbed together, that one which 
stands earlier in the list carries the positive charge, the 
other the negative: 

Fur, The hand, 

Wool, Wood, 

Ivory, Shellac, 

Feathers, Metals, 

Quartz, Resin, 

Glass, Sulphur, 

Cotton, India rubber, 

Linen, Gutta-percha, 

Silk, Celluloid. 

Electrification occurs whenever two bodies not exactly 
similar in both chemical constitution and molecular ar- 
rangement are in contact. The condition of the surface 
of the substances exercises an influence over their electrical 
relations. For this reason plain and ground glass may 
be made to show slight electrical differences. 



8 STATIC ELECTRICITY. 

7. V(olta's Series. — It is generally believed that it is 
not the friction of rubbing that causes electrification, but 
the increased intimacy of contact that is caused by the 
motion. In other words, electrification is due to contact 
of dissimilar substances. Volta showed that contact of 
different metals produced electrification by using a con- 
densing electroscope, an apparatus for detecting very faint 
charges, which is described in section 27. He arranged 
a series, similar to the electrostatic series, but of the metals 
only. As before, each metal is positively electrified by 
contact with any of those following it in the list: 

Zinc, Copper, 

Lead, Silver, 

Tin, Gold. 
Iron, 

More recent investigations have put sodium and mag- 
nesium at the head of the list and platinum and carbon 
(graphite) at the end. 

It must be borne in mind that this series is only true 
for contact of metals in air, since in other atmospheres a 
different order is needed. For instance, in air iron is 
positive to copper, yet in an atmosphere of hydrogen sul- 
phid iron is negative to copper. 

8. Conductors and Insulators. — While all bodies can 
be electrified, only certain of them offer indications of 
electrification in the way of attracting pith balls or re- 
polling similarly electrified bodies. The reason is that 
in some substances the charge remains where generated, 
while in others it disseminates to all parts of the body 
immediately and has not sufficient intensity to give such 
indications. Substances which allow of the dissemination 



GENERAL PHENOMENA. 9 

of the charge are called conductors, while those on which 
the charge remains where generated are called insulators 
or non-conductors. These conditions are relative, not ab- 
solute. Both copper and iron are called conductors, yet 
copper is from 6 to 8 times as good a conductor as iron. 
Dry wood is an insulator to the current from an ordi- 
nary galvanic cell, but is a conductor to the higher ten- 
sions of electrostatic phenomena, and cannot replace the 
vulcanite or glass rods in the various experiments al- 
ready described. Pure water is a very good insulator, 
bnt ordinary water, holding C0 2 in solution, is a rather 
good conductor. The following list of substances, accord- 
ing to Prof. S. P. Thompson, is arranged in the descend- 
ing order of their conductivities: 

Silver, Porcelain, 

Copper, Wool, 

Other metals, Silk, 

Carbon, Resin, 

Water, Gutta-percha, 

The body, Shellac, 

Cotton, Ebonite, 

Dry wood, Paraffin, 

Marble, Glass, 

Paper, Pused quartz. 

Oils, Dry air. 

In considering the choice of a conductor or an insulator 
for any given purpose, not only must its position in this 
series be regarded, but its mechanical and chemical prop- 
erties must be taken into account as well. For conductors 
copper and iron are the two substances most generally 
used. Silver is a better conductor than either, but is too 



•10 STATIC ELECTRICITY. 

^expensive for practical use. Copper is more expensive 
than iron but is so much better a conductor that its extra 
cost is more than compensated for. Iron, besides being 
cheaper, has the additional advantage of greater tensile 
strength, which is an important advantage where wires 
are strung on poles. Carbon is used as a conductor in 
both incandescent and arc lamps, and mercury, the best 
known fluid conductor, is, because of its convenience, 
much used in electrical laboratory instruments. 

The proper choice of an insulator for any particular 
purpose is not a simple matter. Hard rubber, ebonite, 
or vulcanite, is much used and has many advantages in 
point of strength and ease of working. But it is open 
to the serious objection that it looses its insulating quality 
when exposed to light. This is because the sulphur used 
in the process of vulcanizing oxidizes in the presence of 
light. The sulphur oxid then takes up water from the 
atmosphere, forming sulphuric and sulphurous acids. 
These acids, which may be easily tasted by applying the 
tongue to a piece of the rubber, are good conductors, and 
hence as an insulator the hard rubber defeats its own 
purpose. The brown, dull appearance of a piece of hard 
rubber that originally was black and highly polished, is 
due to the destroying action of these acids. A partial 
remedy for this trouble is had by washing the surface 
well with a weak alkali solution, ammonia, for instance, 
then with clear water, and finally rubbing it well with 
paraffin oil. This does not entirely obviate the trouble, 
since the oxidization occurs to some extent beneath the 
surface, where the alkali wash does not reach. Another 
very serious objection to hard rubber is its extraordinarily 
high coefficient of expansion, which is more than twice 
that of glass. This fact alone prevents its use in many 



GENERAL PHENOMENA. 11 

cases where the different parts of an instrument must 
maintain their same relative positions through a consider- 
able range of temperatures. 

Glass is widely used as an insulator. Its chief objec- 
tions are its brittleness and the difficulty of working it. 
It is also very easily wet by water, hence a single drop 
of moisture will spread over a large surface of the glas3 
and materially increase the " surface leakage." 

Quartz, next to dry air, is the best of insulators. It has 
no elastic fatigue, and this quality makes it the very best 
substance where an insulating, flexible, supporting fiber 
is desired. Furthermore it does not wet all over, but what 
moisture settles on it collects into little globules like mer- 
cury on a dusty table. The chief objection to it is the 
great difficulty of working it and the consequent expense. 

Paraffin, called ozokerite, is a good soft insulator, but 
it collects dust, the dust in turn absorbs moisture from 
the atmosphere and surface leakage then occurs. Paraffin 
also has a large coefficient of expansion, but in so plastic 
a substance, this of little importance. 

Mica is a good insulator and cannot be excelled where 
a thin, strong, and rigid layer of insulation is required. 
Its peculiar structure, however, prohibits its use in almost 
all other cases. Mica scales or chips are mixed with oil, 
shellac, rubber, or other materials, making somewhat 
flexible insulating sheets sold under various trade names. 

Kerosene oil is a good insulator, but is carbonized by 
the passage of an electric spark or by the " silent " or 
" brush " discharge, and when carbonized its insulating 
qualities are impaired. Animal and vegetable oils are 
apt to contain organic acids which are more or less con- 
ductors, and in any case such oils are liable to decomposi- 
tion which would render them conductors. 



12 



STATIC ELECTRICITY. 



Shellac is a good insulator when properly prepared. It 
should be dissolved in good alcohol and kept from accumu- 
lating dirt while in its liquid state. Shellac made up 
with wood alcohol contains an organic acid which not only 
impairs its insulation but attacks many metals with which 
it may come in contact. 

9. Influence. — A body may become charged without 
either rubbing or contact with another charged body, 




Fig. 3. 



through the influence of another charged body in its 
neighborhood. This is shown in the following experiment. 
Let the two similar metal hemispheres, Fig. 3, supported 
on insulating stands, be placed with their faces in contact. 
Each hemisphere has a short wire projecting from it, from 
which is suspended by an insulating thread a small pith 
ball. Now let a vulcanite rod be briskly rubbed with a 
woolen cloth and brought near the composite sphere. We 
now find that the point of the sphere nearest the rod, and 



GENERAL PHENOMENA. 13 

the point farthest from it have the power of attracting 
small bits of paper. We see also that the pith balls are 
repelled. Now by testing with a body of a known elec- 
trification we find the ball on the side of the sphere near- 
est the vulcanite rod is positively charged, while that on 
the far side of the sphere is negatively charged. Now 
since they are repelled from the sphere, the latter must 
have a positive charge on the near side and a negative 
charge on the far side. This is evidently due to the in- 
fluence of the vulcanite rod, for the whole phenomena 
ceases on its removal. To clearly understand this phe- 
nomenon it is convenient to adopt the " two fluid " theory, 
according to which every material body contains equal 
amounts of positive and of negative electricity, and on 
electrifying such a body the fluids are separated, the pre- 
ponderance of one or the other fluid giving rise to the 
phenomena which are attributed to charge. Thus on rub- 
bing the vulcanite rod with the woolen cloth the two elec- 
tricities in each of them were separated, the negative col- 
lecting on the rod, while the positive passed over to the 
cloth. Now, when the rod was brought near the sphere, 
the two electricities of the latter were separated, the 
negative electricity of the rod attracting the positive elec- 
tricity of the sphere while it repelled the negative of the 
sphere, hence two charges rested on the sphere, a positive 
charge near the rod, a negative one opposite it. This ac- 
tion it will be observed does not draw from the charge on 
the rod, and the two " fluids " will mix immediately on 
the removal of the rod. If now, while the rod is still near, 
the farther hemisphere be removed from its mate there will 
be a surplus of negative electricity on it which cannot be 
neutralized, when the rod is removed, by the positive 
" fluid " on the other hemisphere because of the insulating 



14 STATIC ELECTRICITY. 

air between. Therefore each hemisphere will carry a 
charge although they are good conductors and could not be 
successfully electrified by rubbing. This phenomenon of 
influence is sometimes called induction. This is unfortu- 
nate, as induction is more widely used as meaning the 
generation of electromotive force in a conductor cutting 
magnetic lines of force; and to distinguish the two, the 
unwieldly expressions electromagnetic induction and elec- 
trostatic induction must be used. So throughout this 
work influence and influenced will be used to the exclusion 
of electrostatic induction and electrostatically induced. 

10. Bound and Free Charges. — If in a conductor a 
charge is held at some particular point through the influ- 
ence of a neighboring charge, the charge so held is said 
to be bound. For every bound charge on a conductor 
there must be an equal charge of opposite kind in a remote 
part of that same conductor or a conductor connected with 
it (provided the continuity of such conductors is not dis- 
turbed during the experiment and that they were not 
already carrying free charges). If there is an actual sur- 
plus of one kind of electricity on a conductor and if it 
is maintained on that conductor simply because the latter 
is completely insulated and the charge has no conducting 
path to lead it away, then such charge is said to be free. 
There can be no free charges of opposite sign in the same 
conductor at the same time. For if there were two such 
charges they would immediately flow together. If they 
were equal they would completely neutralize each other, 
leaving the conductor uncharged, and if they were unequal 
the smaller would neutralize an equal part of the larger, 
leaving the conductor with a diminished charge of the 
same sign as the larger. For example, in the experiment 
cited in the last section, where the hemispheres were 



GENERAL PHENOMENA. 15 

together and the charged vulcanite rod was approached, 
a positive charge was bound on the nearer side of the 
sphere. When the hemispheres were separated and the 
rod removed, the charges distributed themselves over their 
respective conductors and the nearer hemisphere held a 
free positive charge and the farther a free negative charge. 

Of course a bound charge may be superposed on a con- 
ductor already carrying a free charge. In this case the 
free charge is attached or repelled according to its sign, 
and adds itself to the bound charge or to its opposite which 
is located at some remote part of the conductor. In this 
case the bound charge does not have an equal and oppo- 
site charge in the same conductor, the difference being 
just the amount of the original free charge. 

A single free charge can be put on a non-separable sphere 
by proceeding thus : First approach the charged vulcanite 
rod, then touch the sphere with the fingers. The negative 
" fluid " of the sphere in trying to get as far as possible 
from that on the rod will flow through the body into the 
ground. It is quite immaterial what part of the sphere 
is touched. Even if the fingers be placed on that part 
nearest the rod, the positive charge will still be held bound 
and the negative repelled to the earth. 

Now remove the fingers to prevent the return of the 
negative electricity and then remove the rod and a sur- 
plus of positive electricity is on the sphere which cannot 
disperse or neutralize itself because surrounded by insu- 
lators. In other words, the sphere carries a free positive 
charge. 

An application of this principle is found in the electro- 
phorus, illustrated in 'Fig. 4. A circular disc of hard 
rubber, sulphur, or a mixture of shellac, resin, and Venice 
turpentine is suitably held in a metal dish. A metal 



16 



STATIC ELECTRICITY. 



fitted with a glass handle, and of a somewhat 
smaller diameter than the disc, rests upon the latter. If 
now the disc be briskly rubbed with wool, or better with 
cat's fur, it becomes negatively electrified. E"ow plac- 
ing the cover on the disc, by influence a positive charge is 
produced on its under side and a negative charge on top. 
On touching the upper side with the hand the negative 
" fluid " flows to the ground. On removing the hand there 
is a quantity of positive electricity bound on the under side 




Fig. 4. 



of the cover that becomes a free charge when this conduc- 
tor is lifted by the insulating handle. Now on approach- 
ing the cover to the knuckle or other conductor connected 
to the ground, the positive charge attracts a corresponding 
quantity of negative electricity to the knuckle. When the 
distance is sufficiently reduced the attraction between 
these two charges becomes so great that the electricity 
jumps through the air in the form of a spark and a redis- 
tribution of the fluids takes place such that the cover is 
once more neutral or, as it is called, discharged. The cover 



GMBBAL PHENOMENA. 



17 



may be laid on the disc again and the whole operation re- 
peated a number of times until the charge on the disc be- 
comes too weak because of leakage, as it eventually will, 
since neither the air nor substance of the disc is a perfect 
insulator. 

11. Electroscopes. — 'Any instrument for detecting the 
presence of electrification and determining the kind of 
electrification is called an electroscope. Hence the pith 
ball on a string is a crude form of electroscope. A some- 
what better instrument is made by mounting a long thin 




Fig. 5. 




Fig. 6. 



and very light rod of almost any material, wood, straw, 
brass or other, on a needle point so as to swing freely in 
a horizontal plane like a compass needle, as in Fig. 5. 
This will show the presence of charges far too faint to 
affect a pith ball. 

The gold-leaf electroscope is, However, the most delicate 
and most used of this class of instruments. As seen in 
Fig. 6, a flask or other suitably shaped bottle has its mouth 
corked and the cork penetrated hk a glass tube. Through 
this tube runs a brass or copper/rod connected to a brass 
2 



18 STATIC ELECTRICITY. 

disc above and below to two similar pieces of gold-leaf 
hanging side by side and nnited at their upper edges only. 
On bringing a charged body in the vicinity of the disc 
the leaves will diverge, since through the effect of in- 
fluence, a charge will rest on the disc and a charge of the 
opposite kind will rest as far from it as possible in the 
same conductor, or in the gold leaves, and each of the leaves 
having a similar charge, they will be mutually repelled. 
The following is the procedure to determine the kind of 
electrification that a charged body carries. Let a body be 
charged with a known kind of electricity, say by rubbing 
a rubber rod by a woolen cloth giving a negative charge 
on the rod. Approach this rod to the disc of the electro- 
scope. A positive charge will be influenced on the disc and a 
negative charge in the leaves. Now touch the disc with the 
hand. The negative fluid will flow to the ground. (§ 10.) 
Remove first the hand and then the rod. The leaves will 
diverge less than at first, since the negative charge existed 
in them alone while now an equal positive charge is spread 
over the whole of the conductor. The leaves will, however, 
be separated quite sufficiently. ISTow bring the body whose 
electrification it is desired to test into the vicinity of the 
disc. If the leaves diverge further, the body is positively 
electrified, if they come together some, the body is nega- 
tively electrified. For if the body be positively charged, 
this charge will repel the positive " fluid " in the conduc- 
tor of the electroscope to the leaves, thus increasing the 
intensity of the charge there and consequently increasing 
the divergence. If, however, the body be negatively 
charged, this charge will attract the positive " fluid " of 
the conductor to the disc, thus reducing the intensity of 
the charge in the leaves, and consequently reducing the 
divergence. A free negative charge could be placed on 



GENERAL PHENOMENA. 19 

the conductor of the electroscope by touching the disc with 
the negatively charged rubber rod. It is better, however, 
to charge the instrument by influence as indicated, as in 
this way the amount of charge can be regulated, while 
if a highly charged body be touched directly to the disc, 
the gold leaves may repel each other so violently as to do 
themselves damage. 



CHAPTER II. 

THE ELECTROSTATIC FIELD. 

12. Coulomb's Law.— Coulomb's law is stated as fol- 
lows : The force between two charged bodies, far apart as 
compared with their dimensions, is proportional to the pro- 
duct of the charges, and inversely proportional to the 

oo 
square of the distance between them. Or F-=- *\ 

where q and q' are the magnitudes of the charges on the 
two bodies respectively, and r is the distance between 
them. 

This law deduces directly from experiments. The ap- 
paratus used by Coulomb is known as the torsion balance. 
A pith ball is mounted on a light insulating horizontal 
arm. At the center of gravity of the combination is at- 
tached a vertical supporting fiber of glass or metal. The 
pith ball is charged, and a similarly charged body is 
brought into proximity therewith. The ball is repelled, 
and is maintained at a suitable position — far when com- 
pared to its own diameter — by twisting the supporting 
fiber. The torque, and hence the force between the 
charged bodies, is measured by the angle through which 
the fiber is twisted. Now, to subdivide one of these 
charges, let the pith ball be placed in contact for a moment 
with an exactly equal and similar ball which is electrically 
neutral, i. e. without any charge. The charge previously 
on the first becomes evenly distributed over both, and on 
removing the second, the first is left with just half the 
charge it held at first. To cause the balance to maintain 
the same position as before, it is found that just half the 

30 



THE ELECTROSTATIC FIELD. 21 

twist on the fiber is required as was in the first instance. 
By subdividing either or both of the charges to J, J, J, 
etc., it is found invariably that the force between them 
is directly as their product. By maintaining the charges 
constant but varying the distance between them, it can 
be shown that the force varies inversely as the square of 
the distance. 

These experiments should be carried out either in dry 
air or in vacuo, with no other charges and no electrical 
conductors in the neighborhood. 

13. Quantity. — From the expression F = — ^ ma J be 

derived the electrostatic unit of quantity of electricity. 
The units of F and r are already known, the dyne and the 
centimeter. To get unit conditions let F = 1 and r = 1 
and q — q' y then must q = 1. In other words: The unit 
quantity of electricity is such a quantity that when placed 
in air one centimeter from an equal and similar quantity 
of electricity, it will repel it with a force of one dyne. 

It is only in this definition of quantity that the defini- 
tions of the absolute electrostatic units and of the absolute 
electromagnetic units do not agree. The electromagnetic 
unit of quantity is defined as that quantity which when 
flowing in one second through a conductor one centimeter 
long, bent in the arc of a circle of one centimeter radius, 
exerts a force of one dyne upon a unit magnet pole situated 
at the center. This electromagnetic unit is about thirty 
thousand million times the electrostatic unit. Since all 
the other definitions in the two systems are practically 
alike in wording, and since they all include the unit of 
quantity, it follows that the constant numerical factor that 
connects similar units of the two systems is thirty thou- 
sand million, or more exactly 2.9857 X 10 10 . By repeated 



22 STATIC ELECTRICITY. 

observations it has been found that the velocity of propa- 
gation of light in air is 2.9992 XIO 10 centimeters per 
second, and the striking similarity of these figures is of 
considerable significance in sustaining the ether theory 
of electricity. 

The practical electromagnetic unit of quantity is the 
coulomb, and is one-tenth of the absolute unit. There is 
no corresponding practical unit in the electrostatic nomen- 
clature, nor is any special name given to the absolute unit. 
For further comparison of these units, refer to the 
Appendix. 

14. Surface Density. — As before stated, it is convenient 
to consider an electric charge as a quantity of fluid which 
overlies the charged body. With this idea in mind, the 
surface density at any spot may be considered as the thick- 
ness of the fluid at that spot. Independent of such con- 
vention, the definition is: The surface density at any 
point of a charged body is the number of units quantity 
per unit area at that point. 

A convenient instrument for showing the existence and 
behavior of surface density is shown in Fig. 7. A good 
sized sheet of tinfoil is mounted on an insulating roller 
which is supported so that the foil may hang free. Now 
if a charge of electricity be placed upon the foil it will 
immediately spread, covering the whole surface. Thus 
quite a charge may be imparted without giving any great 
density. If now the tinfoil be wound upon the roller the 
surface will materially decrease, and since — as will be 
shown in § 19 — the charge cannot lie within a conductor, 
the same quantity of electricity must reside on a less sur- 
face, hence the density must be greater. That this con- 
dition exists can easily be shown by the gold-leaf electro- 
scope, which shows only slight electrification when ap- 



THE ELECTROSTATIC FIELD. 23 

proached to the open sheet of foil, but shows much more 
intense electrification when approached to the foil when 
rolled up. Or a spark may be drawn from the cylinder of 
foil which could not be drawn from the sheet. 

If a charge be placed upon a conducting sphere and no 
other charges or conductors be in the vicinity, then the 
charge is uniformly distributed over the surface. If a 




Fig. 7. 



sphere have a radius r, its surface is 4-r 2 , and if a charge 
of q units of electricity be imparted to it, then its surface 

density at any point is rr^« 

This condition of uniform density may be represented 
as in Fig. 8 a, where the dotted line shows the thickness 
of the " fluid " on the surface of the sphere. If, however, 
the body is not spherical then the distribution will not be 



24 



STATIC ELECTRICITY. 



uniform, but the surface density will be greatest where 
the radius of curvature is shortest. For instance, on an 
ovoid the distribution will be about as in Fig. 8 ~b, where 
the density is greatest at the sharpest end. Figs. 8 c and 




"X— ' 



Fig. 8. 

d show the distribution on a disc and cylinder respectively. 
In the case of a point where the radius of curvature is 
practically 0, the density becomes so great, as in Fig. 8 e, 
that a curious phenomenon occurs, called the electric wind. 
The electrification at the point is so great as to electrify 




Fig. 9. 



the air particles in the vicinity and immediately repel 
them. The inflowing air particles are in turn charged 
and repelled, and a stream of air is kept flowing which 
may be plainly felt by the hand, or may be made to dis- 
place a candle flame (Fig. 9). Obviously there is a 



TEE ELECTROSTATIC FIELD. 



25 



constant transfer of charge to the air particles, and the 
effect cannot be maintained unless the point is kept sup- 
plied with electricity, as, for instance, from an electro- 
static machine such as is described later. If an electric 
wind is allowed to blow on an insulating surface, say glass 
or ebonite, such surface becomes charged with the same 
kind of electricity as that on the conductor which is caus- 
ing the wind. The facility with which charges leak from 
points renders it imperative that electrostatic apparatus 
be finished in curves and knobs rather than in angles, 




Fig. 10. 



edges, or points, save where a discharge is expressly de- 
sired. Of course the force of repulsion between the point 
and the air particles is mutual. This is easily shown 
by the " electric fly," a few light brass arms radiating 
horizontally from a hub pivoted on a conducting support, 
the outer extremeties being pointed and bent horizontally 
at right angles, as shown in Fig. 10. If now the support 
be connected by a chain or other convenient conductor to 
one terminal of an electric machine, the mutual repulsion 
between the charge on the points and the charges on the air 



26 STATIC ELECTRICITY. 

particles immediately in front of them causes the appara- 
tus to revolve rapidly in the direction opposite that indi- 
cated by the points. This " reaction-mill " is an analogue 
of the Barker's mill known in hydraulic physics. 

15. The Electrostatic Field.— When a body is charged 
with electricity, a strained condition is believed to exist in 
the ether surrounding the body, and the space in which 
this strained condition is felt is called the electrostatic 
field. The field may be represented by lines of force 




Fig. 11. 

emanating from the charged body. A line of force is a 
line drawn in an electric field such that at any point its 
direction is the direction of the electric intensity (§ 16) 
at that point. The field between two spheres is roughly 
shown in Fig. 11. Such lines may be plotted experi- 
mentally by the use of light pivoted straws (Fig. 5), the 
straws always assuming a position parallel to the lines 
of force near them. A free positive charge of electricity 
when placed on a line of force will move along it, and its 



TEE ELECTROSTATIC FIELD. 27 

direction may be considered the positive direction of that 
line. A free negative charge placed upon the line will 
move along it in the opposite direction, and this may be 
considered the negative direction of the line of force. Since 
a positive charge is repelled by a positively charged body 
the positive direction of all lines of force emanating from 
such a body must be away from it, and for a similar reason 
the positive direction of lines emanating from a negatively 
charged body is toward the body. Hence the arrowheads 
in Fig. 11 point away from the positively charged A 
toward the negatively charged B. 

It may readily be shown that the electrostatic field exists 
in vacuo and is independent of air or other gas for its 
existence. This fact is another link in the chain of evi- 
dence which unites electrical phenomena with the lumin- 
if erous ether. 

16. Intensity. — -The intensiiy of an electrostatic field 
at any point is defined as the force in dynes exerted on a 
free unit positive charge at that point, supposing the con- 
dition of the field to be unaltered by the introduction of 
such free unit charge. Obviously there must be a field 
of unit intensity at any point one centimeter distant from 
an isolated unit quantity of electricity (in air), since by 
definition a unit quantity acts with a force of one dyne 
on another unit quantity at a distance of one centimeter. 

In considering an electrostatic field it is usual to let 
each line of force represent a force of one dyne per square 
centimeter. Hence the number of lines per square centi- 
meter of surface measured perpendicular to such lines is a 
measure of the intensity of the field at that surface. 

There can be no intensity at a point within a conductor, 
for if there were, electricity would be moved until an equi- 
librium was obtained, which equilibrium would be an indi- 



,28 STATIC ELECTRICITY. 

cation that no force tended to transfer electricity, which is 
to say that the intensity would be reduced to zero. 

17. Potential — . To determine the character of a field 
it is necessary to know not only the intensity at a given 
point, but also how the intensity varies in the neighbor- 
hood of that point. For instance, at two given points 
the intensity may be just the same, at one, however, due 
to a small near charge, at the other, due to a large far 
charge. At these two points the fields would be very dif- 
ferent, as is shown in Fig. 12. Let A be one square cen- 
timeter perpendicular to the lines of force emanating from 



Fig. 12. 

a charge q at a distance r. As seen in the figure, let there 
be one line per square centimeter at A, then at any point 
in A the intensity is unity. Let also A 7 be a square centi- 
meter perpendicular to the lines emanating- from a charge 
of 25 q at the distance 5 r. Now since the force between 
two charges varies directly as the product of the charges 
and inversely as the square of the distance between them, 
the force acting on a unit charge at A is exactly equal 
to the force acting on a unit charge at A'. Therefore the 
intensity at A' is the same as the intensity at A, and one 
line per square centimeter must be drawn to A' as shown. 
.It is now readily seen that though the intensities are equal 



THE ELECTROSTATIC FIELD. 29 

the fields are quite different, since a slight displacement 
of A' will be attended with only a slight change in inten- 
sity therein, while a slight displacement of A will cause a 
very marked change of intensity at any point in it. 

All these qualities of the electric field are taken into 
account when the potential of a point in the field is con- 
sidered. The potential at any point in an electric field is 
the work which must be expended on a unit positive 
charge to bring it up to that point from an infinite dis- 
tance. It is evident that if the field were due to a nega- 
tive charge that the work expended would be a minus 
quantity, i. e., the unit positive charge would be attracted 
and would perform work. In this case it is convenient to 
regard a point in the field as having negative potential. 
From the foregoing it follows that the difference of 'poten- 
tial between any two points in an electric field is the work 
that must be performed on a unit charge to move it from 
one point to the other. 

Obviously a zero potential will exist only at an infinite 
distance from every charged body. For practical pur- 
poses the potential of the earth is taken as zero. This 
value of zero does not vary because the earth has practi- 
cally an infinite capacity, hence imparting a charge to it 
raises its potential only an infinitesimal amount. This 
assumption of the earth's potential as the zero is analogous 
to the assumption of the sea-level instead of the center of 
the earth, as the zero in measuring the heights of 
mountains. 

A surface which is the locus of all points of a given 
potential is called an equipotential surface. A line drawn 
in one of these surfaces is an equipotential line. In mov- 
ing a charge along an equipotential surface no work is 
done because the potential does not change. Therefore 



30 STATIC ELECTRICITY. 

there can be no component of the force of the field in the 
direction of the motion. Therefore all equipotential sur- 
faces must cut every line of force they meet per- 
pendicularly. 

When a charge is laid on a conductor and assumes a 
state of equilibrium, the surface of that conductor is an 
equipotential surface. For if it "were not, then repulsion 
between the parts of the same charge would cause some 
of the charge to flow from places of higher (numerically) 
potential to those of lower potential. But this is con- 



s'.^ 




Fig. 13. 

tradictory to the first assumption of equilibrium, hence 
the surface of the conductor is an equipotential surface. 
This is true even if the surface density of the charge on 
the conductor varies at different points due to the shape 
of the conductor or the proximity of other charges. This 
argument will not hold good however of a body made up 
of two or more conducting substances, since the mere con- 
tact of dissimilar substances causes electrification on the 
surfaces of contact, positive on the one, negative on the 
other. Since the surface of a charged homogeneous con- 
ductor is equipotential it follows that all the lines of force 



THE ELECTROSTATIC FIELD. 31 

emanating therefrom are normal to that surface on leav- 
ing it. 

Consider an electric charge of quantity q (Fig. 13). 
At any distance r from it, its force on a unit quantity 

will be -^- dynes. This force resolved into the direction 

of the path AB, at any point a, is -^- cos ft, where ft 

is the angle, abc. Hence the work done in moving a 
unit quantity over the infinitesimal distance ds is 

dw = ■—• cos ft ds. 



i* 



But COB ft = ^L 

ds 



7 dr 

.-. aw — a — «• 



Integrating over the distance AB, 



r >dr 



w . 

r* 



w a 



V'i rj 



By definition the work required to move a unit charge 
from one point to another is a measure of the difference 
of potential between those points. Hence the difference 
of potential between two points distant r t and r 2 from a 

charge q and due to that charge is q I 1. Ifr 2 =oo 

we have, when V represents potential, 



w a 



w = V = q -. 



32 STATIC ELECTRICITY. 

This condition complies with the definition of poten- 
tial, so we see that the potential of any point due to a 
charge is as the quantity of the charge and inversely as 
the distance from it. It is well to note here the difference 
between potential and intensity (§ 16), the condition of 
the latter at any point being expressed by 

l= q -. 

By substituting unit conditions in V = — the definition 

arises : — The unit potential is that due to a unit quantity 
of electricity at a distance of one centimeter. 

As the increase of potential energy of a body moved 
from a lower point to a higher against the force of gravi- 




o-- — 



tation is independent of the path moved through, so is the 
difference of electric potential between two points as 
measured by the work done in moving a unit charge from 
one point to the other independent of the path the charge 
is moved through. 

In Fig. 14 let V be the potential at a point A and 
V + dV that at the indefinitely near point B. Let I be the 
intensity of field at A B and F that part of the force which 
resolves in the direction A B. The work required to move 
a unit of electricity from A to B is the (negative) work 
required to move it from A to an infinite distance plus 
the work required to move it from an infinite distance, 



THE ELECTROSTATIC FIELD. 



33 



back to B, that is, (V + dV) — V. But the work is also 
Fds, 

.:Fds = (V+dV)-V 

ds 

dv 
But F — I cos ft and cos /? — y-, therefore i^fo = I dr, 



so 



/ 



dr ' 



18. Flux Through a Closed Surface. — If in an electric 
field a closed surface be described, if Q be the quantity 




Fig. 15. 



of electricity enclosed and if IV be the component of the 
intensity at any element of surface, ds, resolved normally 
to ds and outward ; then the ff Nds taken over the 
whole surface is 4ttQ. That is, there will emanate 4tt§ 
lines of force from that surface (§16). 

To prove this theorem, consider the enclosed quantity 
Q, Fig. 15, to be concentrated at some point 0. Let r 
3 



34 STATIC ELECTRICITY. 

be the distance of the elementary surface ds from 0. 
Let be the angle between the direction of the intensity, 
7, and the normal component thereof, N. Now at ds, 

I— Hr. Therefore iT = % cos 0. and iV^s = %- cos 0ds. 

Now consider the cone whose apex is at and whose 
elements bound the surface ds. If this cone be given the 
normal base mn at ds, then the area mn = cos ds, 
because one linear dimension of the areas is common, 
and the other linear dimension is in the ratio of 1 : cos 0, 

But 2 — is a measure of the solid angle at the 

apex of the cone. Hence cos e ds is numerically equal to 




Fig. 16. 

the solid angle subtended at by ds. This expression is 
+ or — according to the sign of cos 0. We know that 
this sign must be + when faces the inside of the closed 
surface, and — when it faces the outside. 

Now a right line draw from must cut the surface 
an odd number of times, as, for instance, in Fig. 16. A 
cone such as we have been considering which followed the 

line in Fig. 16 would contribute the term ~~ cos $ ds 

r 

at the first intersection, -^ cos 'ds at the second, 

Ti 



and -£; cos e"ds at the third. 



TEE ELECTROSTATIC FIELD. 35 

^ cos e' ds cos e"ds .,,,,, 

But — - 2 — = 2~ since they both measure the 

same solid angle. So two of these terms balance out, 
and only the first need be considered. Thus, with any de- 
gree of re-entrancy, all other terms cut themselves out 
pair by pair, save the first, which is Q X the solid angle 
of the cone. 

For the integrated value of the force all over the sur- 
face we get Q X the sum of the solid angles of all the cones 
filling the space about 0, or k-rcQ. Therefore, 

//Nds =ff-Q cos ods = ±* Q. 

which was to be proved. 

If the charge Q were at some point 0' outside the sur- 
face, then the intersections for any cone would be of an 
even number, contributing an even number of terms al- 
ternately + and — -£- cos o ds, which would reduce the 

value f/Nds to 0. 

Evidently there must emanate from a unit quantity of 
electricity 4?r lines of force. 

19. There is no Charge Within a Conductor. — The re- 
lation just derived offers means of reaching conclusions 
analytically that may be shown to be correct experimen- 
tally. Thus we find that there can be no electric charge 
in a conductor. For, consider an indefinitely small closed 
surface, described in the body of a charged conductor. 
There is no intensity at this surface, since if there were the 
electricity would be moved, while we know that it is in 
equilibrium. There being no intensity, N must be equal 
to 0. Hence f/Nds = = 4 n Q 

.-. Q = 
and there is no charge within the body of the conductor. 



36 



STATIC ELECTRICITY. 



This fact is shown experimentally by Biot's spheres. 
A conducting sphere (Fig. 17) is suspended by an in- 
sulating thread and a charge imparted to it. Two hollow 
conducting hemispheres nicely fitting the sphere and 
mounted with insulating handles are placed so as to al- 
most surround but not touch the sphere. By means of a 
proof plane — a small conducting disc, or sometimes a 
bead, on an insulating handle, by which a sample of a 
charge may be taken from a large conductor and examined 
in the usual way with an electroscope — the charge can be 
shown to be still present on the sphere. Now close the 




Fig. 17. 



hemispheres around the sphere, making electrically one 
sphere of somewhat greater diameter. On removing the 
hemispheres no trace of charge can be found on the sphere, 
but it can all be found on the two pieces which for a mo- 
ment had formed the whole surface of the conductor. 

A pair of gold leaves hung inside a wire cage cannot be 
made to diverge when the cage is electrified, even if in 
electrical connection therewith. In elaboration of this ex- 
periment Faraday constructed a twelve-foot hollow cube 
of wood, covered it with tinfoil and placed it on insulating 
supports. He charged it so highly that sparks and brush 
discharges issued from the corners and edges, and then 



TEE ELECTROSTATIC FIELD. 37 

placed his most delicate electroscopes inside, but once in- 
side the gold leaves showed no tendency to diverge. 

Faraday also constructed a bag of linen gauze — a fairly 
good conductor to static charges — mounted on an insu- 
lated ring, as seen in Fig. 18, with a silk string so fixed to 
its bottom that it could be readily turned inside out. A 
charge imparted to the bag was always found on the out- 
side, no matter how frequently or suddenly the inside 
might be made the outside. 




Fig. 18. 

20. Charges in a Cavity. — When inside of a conductor 
there is a cavity, and in this cavity is a charge insulated 
from the conductor, then on the inside surface of that 
conductor there will be a charge caused by influence, but 
there is none in the body of the conductor. That the in- 
sulated charge and that induced by it on the inside surface 
of the conductor are exactly equal is shown as follows: 
Imagine a closed surface described within the body of the 
conductor and surrounding the cavity. There is no inten- 
sity at this surface (§ 16), hence N = and ffNds = 
= 4:ttQ, hence Q = 0. That is, the algebraic sum of the 



38 



STATIC ELECTRICITY. 



charges within the surface is 0. Therefore, the insulated 
charge induces an exactly equal charge of opposite sign 
on the inner surface of the conductor. 

Faraday's experimental proof of this is widely known 
as his " ice pail " experiment, since he used a metal ice pail 




Fig. 19. 



in performing it. Let the outside of a metal pail or cyl- 
inder, standing on an insulating block, be connected by a 
wire to an electroscope (Fig. 19). ~No divergence is ob- 
served. Lower a charged ball into the pail by means of a 
silk thread. The leaves will gradually diverge until the 
ball is thoroughly within the pail, when they will assume a 
steady position. Allow the ball to touch the inside of the 
pail. The leaves will not move. Remove the ball entirely. 
The leaves remain at their constant divergence, and the 
ball will be found to have lost all its charge. The explana- 



TEE ELECTROSTATIC FIELD. 39 

tion is as follows. Suppose the ball to be positively 
charged. It influences on the inside of the pail a negative 
charge. This leaves a positive charge on the outside of the 
pail, which is registered by the electroscope. When the 
ball touches the pail its charge and the negative charge in 
the pail neutralize each other. If they were not exactly 
equal the gold leaves will be diverged or contracted as the 
positive or the negative charge was in excess. 

This experiment shows also that there cannot be a 
charge influenced in a body without there being an ex- 
actly equal and opposite charge in the same body or one 
electrically connected therewith, which charges flow to- 
gether and neutralize each other on the removal of the 
influencing charge, provided that the continuity of the 
conductor is not disturbed during the experiment, and that 
no free charges rested upon the conductor before the ex- 
periment. 

21. Tubes of Force. — ■ Imagine a plane in an electric 
field normal to the direction of the electric intensity, and 
a closed figure drawn on such plane. If now a line of 
force be drawn through each point of this closed figure, 
such lines will define a tubular surface. Such a tubular 
surface is called a tube of force. 

Consider a tube of force of indefinitely small section 
(Fig. 20). Let a portion of it be made into a closed 
surface by two normal sections s and s'. Let there be no 
electricity within this surface. Then ff Nds = 0. Let 
the intensity be I at s and V at s'. The whole normal out- 
ward force at s is Is and at s' is — I's' (negative because 
opposite in direction to the lines of force). There can be 
no force from the other sides of this closed surface, since 



40 STATIC ELECTRICITY. 

there can be no component of intensity normal to a line 
of force; therefore 7s + ( — I's') = 0, or Is = I's'. 

By convention we draw as many lines per unit area 
of s as is the value of I .expressed in dynes. But from 
Is — I's' we see that I' is inversely as .s', hence all lines 
of force passing through s will, when produced, pass 
through s'. So the intensity of any point in a field is rep- 
resented by the number of lines of force per normal unit 
of area at that point. 

If equipotential surfaces be drawn, spaced so that there 
is an infinitesimal difference of potential dV between any 




Fig. 20. 

two adjacent surfaces, then the number of these surfaces 
intercepting an infinitesimal length of a line of force, dr, 
is a measure of the intensity there. For, from section 17, 

I— -j-, that is, the intensity at any point is inversely as 

the distance between the equipotential surfaces, or directly 
as the number per unit length. 

As long as there is no electricity in a tube of force the 
product Is remains constant. It is evident that there will 
be a value for / however far the tube may be extended. 
If s = oo , then I becomes zero, and the tube has an end. 
Hence we see that a tube and therefore a line of force 
can begin or end only in a surface charged with electricity. 



THE ELECTROSTATIC FIELD. 41 

Consider a tube of force passing from one charged sur- 
face to another charged surface of the conductors which 
form its ends. Prolong the tubes slightly into the con- 
ductors and close their ends by planes. The whole tube 
is now a closed surface. There can be no intensity at 
those parts of the surface within the conductors, and there 
can be no component of / normal to the tube outside the 
conductors, therefore = Nds = 0, and the charge on the 
surface of one conductor embraced by one end of the tube 
must be exactly equal and of opposite sign to the charge 
on the surface of the other conductor embraced by the 
other end of the tube, to fulfil the condition Q — in 
//Nds = = ^Q: 

22. Intensity Outside a Charged Surface. — Let P (Fig. 
21) be a point on a charged surface whose density is G - 



Fig. 21. 

Pass a tube of force through the charged surface, cutting 
out of it a very small surface s. Close the tube inside the 
conductor by a plane and outside by a surface parallel to 
the charged surface and indefinitely close thereto. Since 
the lines of force are normal to s they are also normal to 
this last surface, hence it is in all respects equal to s. The 
cylinder thus constructed around P forms a closed surface 
which encloses the charge <?s. If I is the intensity just 
outside. of s, the whole normal force issuing from the 
closed surface is Is. Therefore // Nds = 4:ttQ becomes 

Js = 4 - <j s, 

.*. I — 4 iz &. - °« 



42 STATIC ELECTRICITY. 

The electric intensity just outside a conductor whose sur- 
face density is ff is 4 tz<t. 

23. Stress on an Electrified Surface. — In Fig. 22, let 
P be a point on a conductor whose surface density is fl- 
at which it is desired to find the surface stress. Consider 
a surface AB about P so small as to be sensibly a plane. 
On the normal at P consider the points m and n, one in- 
side, the other outside the conductor, and equally distant 
from P. Now m and n are so indefinitely near together as 



i 



Fig. 22. 

to be sensibly one point to all parts of the charge, save 
that lying on AB. The intensity all over AB, being nor- 
mal thereto (§ 17), is parallel to Pm. Consider the 
forces acting at m and n. They may be divided into the 
force F, due to the portion of the charge on AB, and the 
force F' due to the rest of the charge. Due to the charge 
on AB, a unit charge at m would be urged in the + di- 
rection with a force F, while at n it would be urged in the 
— direction with a force F. Due to the rest of the charge, 
at either m or n, these being now sensibly one point, it 
would be urged in the + direction with a force F' . Hence 
the total force, or the intensity, at m is F' + F , while that 
at n is F' — F. But we know that the intensity at m 
is 4 7r<r(§ 22), while that at n is (§ 19). Hence 

F + F = 4 n* 

F'-F = 



TEE ELECTROSTATIC FIELD. 



43 



That is, the rest of the charge exerts a force of 2 tt<t per 
unit charge on A B, and in the direction P m. When 
<f and s are measured in the absolute units, the total stress 
on the surface is normal thereto and of a force of 2 n ff x as 
dynes. Or the normal outward stress is 2^ dynes per 
square centimeter. 

This stress may be expressed in terms of / just outside 



the surface, for 



Stress per unit area = 2^ 
.". Stress per unit area = -x~ I 2 , 



22), 




The effect of this stress can be qualitatively shown by 
blowing a soap bubble on a metal pipe. After closing the 
aperture and holding by means of some insulator, a charge 
may be imparted to the pipe which is immediately coin- 



44 STATIC ELECTRICITY. 

municated to the bubble. The bubble will be seen to ex- 
pand, the electric repulsion or surface stress overcoming 
a part of the molecular cohesion. 

It is stated that a surface stress of 66,700 dynes per 
square centimeter is sufficient to cause the electricity to 
leave the surface, passing through the air in the form of 
a spark. 

24. Attraction Due to an Electrified Plane. — If there be 
an indefinitely extended plane, a portion of which is rep- 
resented in Fig. 23, charged to a surface density <t, and if 
a quantity of electricity Q be at the vertical distance a 
from that plane, the force of attraction (or repulsion) 
exerted on that quantity by the plane may be determined 
as follows : 

The attraction between any infinitesimal quantity dq 
on the plane, for the other quantity Q is (§ 12) 

dF' = SJf 

/pa 

where r is the distance between the quantities. The com- 
ponent dF of this force which tends to attract Q to the 
plane is 

dF = dF' cos fi 

= ^ f COS J3. 

But dq = <r dS 

and taking a circumferential differential of area, 

dS = 2 7T x dx 
,\ dq = 2 7i <t x dx. 
Also r 2 = a 2 + ^ 2 

and cos ft = ■ , 

Va 2 + x 2 



TEE ELECTROSTATIC FIELD. 45 

Hence 



dF = 



2 7T (T a Q x d x 



(a 2 + x>)Va;* + x 2 
and as the plane is indefinitely extended, 

F=27t<raQ r 

= 2 7T ff (). 

From this it is seen that the attraction exerted on a unit 
quantity of electricity by an indefinitely extended plane 
charged to a surface density <? is 2 ?r <r dynes, and is inde- 
pendent of the distance between the unit quantity and the 
plane. 

If a unit quantity of electricity be placed between two 
indefinitely extended parallel plates equally and oppositely 
charged, as for instance ordinary condenser plates, the 
surface density on each will be a. One plate will attract 
the unit quantity with a force of 2 x a dynes, the other will 
repel it with a force of 2 tt a- dynes. Therefore the re- 
sultant force on it will be 4 tt<7 dynes. If V represent 
the potential of one plate and V that of the other, then 
their difference of potential is V — V. But from section 
17 their difference of potential is measured by the number 
of ergs of work necessary to move a unit quantity from one 
to the other. Work = force X distance, hence 

V— V = ±izad 
where d is the distance between the plates. 



CHAPTER III. 

CAPACITY. 

25. Definition. — The electrostatic capacity of a con- 
ductor is measured by the quantity of electricity which 
must be imparted to it to raise its potential from zero to 
unity. Thus the capacity of a conductor is not the quan- 
tity it can hold without running over, as a quart measure, 
nor is it the amount it will hold without bursting or dis- 
charging or in any way taxing itself to a limit. 

The most generally used unit of capacity is the farad. 
This unit belongs to the electromagnetic system, i. e., & con- 
ductor in which one coulomb (ampere-second) will raise 
the pressure one volt has one farad capacity. Practical 
capacities are vastly smaller than a farad, so the term 
microfarad ( = .000001 farad) is employed. To reduce 
a capacity expressed in c. g. s. electrostatic units to farads, 
divide by 9 x 10 11 , or what is of more utility, to reduce a 
capacity in electrostatic units to microfarads, divide by 
900,000. This constant is not an exact multiple of 10 be- 
cause of the difference existing between the electrostatic 
and electromagnetic systems, which has its origin in the 
different definitions of a unit quantity. (See § 13.) 
These relations are discussed more fully in the Appendix. 

26. Capacity of a Sphere. — « If a sphere of one centi- 
meter radius be charged with a unit quantity of electricity, 
the potential at any point on the surface will be raised to 
unity. For the charge may be considered as concen- 
trated at the center of the sphere, and by definition (§ 17), 

46 



CAPACITY. 



47 



there is unit potential at a point one centimeter from a 
unit charge. Hence the sphere has a unit electrostatic 
capacity. Also if the sphere have a radius r, then it 
must be charged with r units of electricity to raise its 
surface potential to unity, and its capacity is equal to its 
radius expressed in centimeters. 

Considering the earth a perfect sphere of 4,000 miles 
radius, its capacity is 

4000 x 5280x32x2,54 



900000 



= 715 microfarads. 





Fig. 24. 



27. Principle of the Condenser. — The capacity of any 
conductor is not dependent solely upon that conductor, but 
also upon the nature of its surroundings. The relation 
deduced in the last section is true only for a sphere iso- 
lated in air. Consider two conducting discs (Fig. 24) 
on insulating supports and with rough pith-ball electro- 
scopes, as shown. If the knob a be connected to the neg- 
ative terminal of an electric machine, the disc A will be- 
come negatively charged, and will influence an equal 
positive charge on the disc B. The negative charge re- 
pelled to knob b may then be disposed of by touching b 



48 STATIC ELECTRICITY. 

with a grounded wire, and the connection between the 
electric machine and a may then be severed. We now 
have equal and opposite charges or quantities on the two 
conductors. The potentials at all points of a conductor 
must be the same (§17) and the pith-balls will indicate 
the changes of potential of the conductors by their greater 
or less repulsion. 

If now the discs A and B be separated, the attraction 
between the charges will be less, and some of the bound 
charges will become free, causing an increased indication 
of the pith-balls; that is, the potential of the conductors 
has increased. It follows from the definition of capacity 
that since the potential rises and we know the total quan- 
tity has remained constant, the capacity of either of the 
conductors must have diminished. If A were moved to 
an infinite distance, the potential of B would be a maxi- 
mum for the given quantity and its capacity a minimum, 
the case of an isolated conductor. 

If this experiment be repeated with some dielectric other 
than air, glass for instance, between A and B, and more 
accurate means of measuring the potential be employed, 
it can be shown that the potential rise for a given charge 
and a given distance between the conductors is different 
than it was with air. This difference is further discussed 
in section 29. 

From these facts it seems that the capacity of such an 
arrangement of conductors and dielectrics depends upon 
three things: (1) Size and shape of conductors; (2) 
thickness of dielectric between them; and (3) nature of 
dielectric. 

A useful application of the principle that the potential 



CAPACITY. 49 

of one conductor of a charged condenser increases when the 
conductors are separated is in the condensing electroscope 
referred to in section 7. In Fig. 6 is shown an ordinary 
gold leaf electroscope the conductor of which terminates in 
a metal disc. If now a similar disc, supplied with an in- 
sulating handle, and having its surface well covered with 
varnish, shellac or lacquer, be placed upon the first disc, 
the two will form a condenser, the varnish being the dielec- 
tric, "Now a charge of so low a potential as to give no 
indication with the ordinary electroscope can be examined 
by imparting some of the charge to the lower disc while 
the upper disc is grounded by touching with the finger or 
otherwise. Because of the condenser action of the plates 
the low potential can impart a relatively large amount of 
charge to the disc. The source of original charge should 
first be removed and then the upper disc taken away, thus 
greatly lowering the capacity of the lower plate, and as the 
charge on it has no means of escape its potential is greatly 
increased and gives a satisfactory divergence of the gold 
leaves. 

28. Leyden Jar. — Musschenbrock and Cuneus in the 
city of Leyden were trying to get a bottle full of electric 
" fluid." The bottle was half full of water and the " fluid"" 
was to be led in by a long nail extending through the cork 
into the water. While being charged the bottle was held in 
the hand. The water then was one conductor, the hand 
the other and the glass the dielectric between, fulfilling 
the conditions of the experiment of the last section. On 
touching the nail with the free hand Cuneus discharged 
the condenser, getting a " shock." 

4 



50 



STATIC ELECTRICITY. 



From this beginning the Ley den jar has developed to its 
present form, usually a glass jar, Fig. 25, coated on its 
bottom and about two-thirds of its height, both inside and 
outside, with tinfoil. A loose fitting wooden cork carries 
a brass rod terminating at the top in a knob and at the 
bottom in a bit of brass chain which gives contact with the 




Fig. 25. 



inner coating. To charge, the knob is held against the 
terminal of an electric machine and the outer coating is 
grounded, either by the hand or a grounded connection. 
To discharge, a knuckle of the free hand may be presented 
to the knob giving a spark and a " shock ;" or better, the 
discharging tongs, movable metal legs terminating in 
knobs and having one or two insulating handles, Fig. 26, 



CAPACITY. 51 

may be laid across the outer coating and the knob. In 
nsing the tongs contact should first be made to the coating 
and then the other leg presented to the knob. If the re- 
verse order is followed the spark will be drawn from the 
outer coating and may damage it. 

A Leyden jar whose diameter is 23 cm. and the coating 
33 cm. high has about 2,700 sq. cm. area for each coating. 
If the glass be .2 to .3 cm. thick, the capacity of the jar will 
be .004 to .008 microfarads. A condenser consisting of 
two circular, coaxal, metallic plates, 20 cm. in diameter 




Fig. 26. 

and 1 cm. thick, using dry air as the dielectric will have a 
capacity depending upon the distance between them as fol- 
lows: 

.1 cm. .000289 microfarad. 

.2 cm. .000149 microfarad. 

.5 cm. .000064 microfarad. 
1.0 cm. .000035 microfarad. 



£2 STATIC ELECTRICITY. 

29. Seat of the Charge. — If a Ley den jar be built up of 
a metal cup, a glass cup and another metal cup, each 
fitting snugly into the next, the metal cups replace the 
tinfoil of the ordinary apparatus and may be removed 
for experiment. If such a jar be charged and then dis- 
assembled without discharging, the metal parts show little 
or no charge. The jar may be reassembled, with other 
metal cups if desired, and the whole is still found to be 
charged. This shows that the seat of the charge is on the 
surface of the glass. The charge on one side being nega- 
tive and that on the other side positive, they attract each 
other and hold themselves in place, squeezing the glass 
between them. This is a real mechanical stress on the 
glass and may become so severe as to break or pierce it, 
an accident not uncommon with Ley den jars and which 
ruins them completely. The utility of the metal coatings 
would seem to be only to distribute the charge over the 
glass, or, on discharging, to convey the charge instantly 
from all over the glass to the point of discharge. 

30. Specific Inductive Capacity. — Cavendish first dis- 
covered that the capacity of a condenser depended on the 
dielectric used as well as on its dimensions and arrange- 
ment, but Faraday carried the work further and gave 
quantitative results. If two condensers are exactly similar 
save that air is the dielectric in one and the substance 
under consideration is the dielectric of the other, then the 
ratio of the capacity of the second condenser divided by 
the capacity of the air condenser is called the specific 
inductive capacity of the dielectric. If the discs of Fig. 
24 are immersed in kerosene their capacity will be double 
that which they have in air. Therefore kerosene has a 
specific inductive capacity of two (2.03-2.07). The terms 



CAPACITY. 53 

dielectric coefficient and inductivity have been employed 
in this sense. The latter, though little used, seems to be 
the best because of its brevity and its analogy to conduc- 
tivity as applied to the non-insulators. 

While all dielectrics are good insulators, it is not equally 
true that all insulators are good dielectrics. For instance, 
paraffin is an excellent insulator, yet its specific inductive 
capacity is but two, while that of mica is six or eight, and 
that of glass anywhere from three to ten, according to the 
observer. There are two reasons for the wide discrepancy 
in the results of different experimenters on the specific in- 
ductive capacity of glass apart from a difference in the 
physical constitution of the glasses tested. First, the 
dielectric seems to absorb part of the charge as time goes 
on, thus increasing the capacity of the condenser. This 
phenomenon of " elastic fatigue " is treated in the next 
section. Second, since nothing is an absolute insulator, 
there is a constant conductive discharge in an electrified 
condenser. Because of this continuous leak, a method 
which measures the capacity at the moment of charge will 
give a much higher " inductivity " for a given dielectric 
than will a method that requires an appreciable space of 
time between the charging and the measurement. The 
specific inductive capacity of nearly perfect vacuum is 
.999. In determining the specific inductive capacity con- 
stant of a dielectric by comparison with an air condenser, 
the air should be perfectly dry, of a temperature of 0° C. 
and under a barometric pressure of 760 mm. 

The following table gives approximately the specific in- 
ductive capacities of the more generally used dielectrics : 

Air 1 

Glass 3 to 7 



54 STATIC ELECTRICITY. 

Ebonite 2.2 to 3 

Gutta-percha . . . ., 2.5 

Paraffin 2 to 2.3 

Shellac ., 2.75 

Mica Q.Q 

Beeswax ,. 1.8 

Kerosene . . . 2 to 2.7 

31. Mechanical Stress Due to Electrostatic Charge. — »It is 
very convenient to consider dielectric stresses as exactly 
analogous to elastic stresses in other bodies. This hypoth- 
esis will explain all the mechanical effects noted in 
dielectrics, residual charge, electric expansion, and dielec- 
tric " elastic limit," each of which will be explained. 

The first mechanical effect noticed is the breaking down 
of the dielectric when a condenser is overcharged. As 
when a spark is propelled across an air gap, so when a 
hole is pierced through the glass of a Ley den jar, there 
must have been real mechanical energy expended in the 
phenomenon. In such a case the attraction of the charges 
on the opposite sides of the dielectric strained it so severely 
that it broke, the same as a drum head, under too heavy a 
pressure, will rupture. 

When a Ley den jar has been charged and discharged, 
after the lapse of some minutes it may be discharged 
again, much more feebly but yet noticeably. This 
phenomenon may be repeated two or three times, each time 
more feebly and with longer waits between. On charging 
the jar the molecules of the glass were squeezed together 
by the attraction between the charges on the opposite sides. 
Glass is not perfectly elastic, so on discharging the mole- 
cules did not at once resume their normal position, but 



CAPACITY. 55 

took some position approximating the normal. After a 
lapse of time they are prepared to resume more nearly their 
normal position and a further discharge enables them to 
do so. Air is nearly perfectly elastic, and an air condenser 
has no detectable residual charge, A glass fibre when 
twisted will return almost to normal instantly, and reach 
its normal only after the lapse of time. A quartz fibre has 
no such elastic fatigue, but returns all the way to normal 
at once. A glass condenser has a noticeable residual charge 
while a condenser which has layers of quartz crystal for 
its dielectric shows no residual charge. 

" Electric expansion " is the term given to the phenom- 
enon exhibited by the Leyden jar of expanding when 
charged so that its cubic contents is increased. Duter 
claimed to have shown that this expansion is inversely 
proportional to the thickness of the glass, and also pro- 
portional to the square of the potential difference. If we 
consider the glass to be subjected to a squeeze along one of 
its dimensions, i. e., its thickness, it is natural to expect it 
to expand in its other two directions, that is, in the height 
and in the circumference of the jar. This was the ex- 
planation adapted by Priestly and Volta. 

Recent experiments go to show that there is a limit to a 
dielectric analogous to the elastic limit of metals. If a 
steel bar be stressed to any point within a certain limit it 
will on being set free resume its original condition, while 
if the stress is carried beyond this point the bar takes a 
permanent set and its elasticity and its ultimate strength 
are seriously impaired. So also with dielectrics. When 
subjected to moderate potentials they will return to their 
original condition, but when the stress is carried beyond 
the " elastic limit " the insulator takes a permanent " set " 



56 STATIC ELECTRICITY. 

and its future usefulness is impaired. Thus it is quite 
possible that in testing insulators their value may be 
seriously reduced by subjecting them to a voltage above 
their " elastic limit " but below their ultimate break-down 
point, and the injury not be detected till they are put 
in service. 

32. Dielectric Hysteresis. — When a condenser is charged 
there is a loss of energy in the dielectric that appears as 
heat. A small portion of this loss is due to the leakage of 
charge through the resistance of the dielectric. This pas- 
sage of electricity is really a current and thus classes the 
phenomenon as electrodynamic. The energy appearing as 
heat may be represented as I 2 R watts, where I is the cur- 
rent in amperes and B the resistance in ohms. This loss 
is practically infinitesimal. A much greater loss of energy 
is due to what seems to be a molecular friction in the body 
of the insulator, which is quite analogous to the magnetic 
hysteresis of iron. A physical conception of this action 
may be had by considering that the dielectric is made up 
of a number of small particles, adjacent but not touching, 
which we may call molecules. Each molecule has a small 
positive charge bound at one end and an equal negative 
charge at the other. These charges in their turn bind other 
charges on adjacent molecules. The molecules are irregu- 
larly disposed and the extremities carrying the positive 
charges may point in all directions indiscriminately. Now 
when a positive charge is laid on one plate of the condenser 
and a negative charge on the other, these molecules are all 
swung around by the attraction between unlike charges so 
that their negative charges will all point toward the posi- 
tively charged plate. The work thus expended to swing 
them is not stored either potentially or kinetically but ap- 



CAPACITY. 57 

pears as heat. This work is the loss due to dielectric hys- 
teresis. The student must remember that this reasoning is 
hypothetical and is introduced as are analogies in other 
parts of the work, merely to aid in giving a firmer grasp 
on the subject. There is no independent proof of the ex- 
istence of such charges on the molecules, nor indeed is there 
any such proof of the existence of the molecules them- 
selves. 

The phenomenon of dielectric hysteresis while of some 
importance in electrodynamics has but slight influence in 
electrostatics, but it is treated here as being an essential 
part of the study of the condenser. The existence of this 
phenomenon is easily shown. For instance, a paraffin con- 
denser whose ohmic resistance is, say, one megohm will 
when connected to a source of E.M.F. of 100 volts suffer 
an I 2 B loss in the dielectric thus : 

E lo 6 
BR = 10- 8 X 10 6 = lO" 2 watt. 

The temperature rise due to this loss would be so small 
as to be beyond the range of ordinary measurement. Yet 
if this source of E.M.F. be' of reasonably high frequency, 
125 ~ (125 complete cycles per second), for instance, the 
condenser may soon get so hot that £he paraffin will melt 
and run out. If air bubbles had been left in the dielectric 
in the process of manufacture, the bombarding of the air 
particles back and forth due to their alternate attraction 
and repulsion might account for some of the rise of tem- 
perature, but this in itself is not of sufficient magnitude to 
explain the phenomenon. 

We have already seen that the best insulators did not 



58 STATIC ELECTRICITY. 

necessarily make the best dielectrics for condensers, be- 
cause of the difference in their specific inductive capacities, 
and in dielectric hysteresis we have another factor to con- 
sider in selecting a material for such a purpose. Air is a 
better insulator in the ohmic sense than either oil or mica, 
but it is inferior to these in specific inductive capacity, in 
disruptive strength, and in dielectric hysteresis, hence 
either of the latter is preferable to the former for such use. 

33. Calculation of Capacity. — It is almost impossible to 
calculate directly the capacity of ordinary conductors, but 
a few of the simpler geometric forms can be treated 
theoretically. 

If G = the capacity of a condenser, V = the difference 
of potential between the conductors, and Q = the quantity 
with which it is charged, then by definition of capacity 

a- Q 

o- T . 

(a). If we have two concentric conducting spheres (Fig. 
27), the smaller of radius r, the larger being hollow and of 




Fig. 27. 



internal radius R, and a charge Q rest upon the smaller, 
then a charge Q will be influenced on the inside of the 
larger. By section 19 the potential at any point within a 
closed conductor is the same as that of the conductor, and 



CAPACITY. 59 



by section 17, the potential at the common center of the 
spheres is to the algebraic sum of each of the neighboring 
charges divided by its distance or 

Fat center = -^ — -%- 
r R 

and this is the potential of the inner sphere. Since there 

is no free charge on the outer sphere its potential is zero 

and the difference of potential between the conductors is 

V - r R ~ V \ r R } 

Substituting this in our capacity equation we have 

_£__ 1 rE 



V "~ _1_ _1_ ~ R-r 
r R 
The capacity of two concentric spheres is equal to the 
product of their radii divided by the difference of their 
radii. 

If instead of air some dielectric whose specific induc- 
tive capacity is h be employed, the equation becomes 

n 7 r R 

R — r 

(b). To find the capacity of two portions, each of area 
8 and opposite to the other, of two indefinitely extended 
parallel plates, distant a apart. Let the two plates be 
charged with equal and opposite quantities of electricity. 
These quantities will be distributed evenly over the plates. 
Let the quantity per unit area be q. Being indefinitely 
extended plates, the influence of their edges is neglected. 
The potential at any point P on the positively charged 
plate due to the charge on any circumferential differential 

of area (Fig. 28) i$*lLlj!L y (quantity divided by dis- 



60 STATIC ELECTRICITY. 

tance.) Therefore the total potential at P due to the 
positively charged plate is the sum of all such differentials 

/* 2 It 7* a d 7* 
. Since the 
u r 
plates are oppositely charged, and are a distance apart, 




Fig. 28. 

the potential at P due to the negatively charged plate is 

y-oo ^Tvrgdr __ . .. . , 
. We may now write the total potential at 
o V r 2 -\- a 2 

P 

"v o y y— Yr 2 + a 2 ) 

= 2 * ? rll - L \ dr 

*So \ V?> 2 -\-a 2 ) 

= 2rrq I T- \/ T 2 + a 2 

£= 2 7T a q. 



CAPACITY. 61 

Likewise the potential at a corresponding point on the 
negatively charged plate is 

V" = - 2rraq. 

The total difference of potential between the plates is 

The quantity of electricity on the portion 8 of one of 
the plates, Q = Sq. 

Substituting in the capacity equation 

C =_Q ^ Sq _ 8 
V 4 7T a q 4: 7i a' 

If the plates are of finite dimensions but of great dimen- 
sions when measured in terms of the separating distance, 
the same formula will apply without serious error. 

The capacity of two similar plates very near together is 
equal to the area of one side of one plate divided by 4m 
times the distance between them. 

If the separating medium is other than air, the formula 
becomes 

4crr a 
where h is the specific inductive capacity of the dielectric, 
(c). To find the capacity of the portions cut off from 
two indefinitely long coaxal cylinders of radii r and R 
by two planes perpendicular to the axis and distant I apart. 
Suppose a charge be distributed over the inner cylinder 
(Fig. 29) such that there will be a quantity q per unit 
area. By a reasoning analogous to that employed in the 
case of the concentric spheres it can be shown that the 
outer cylinder will be at zero potential and that the poten- 
tial of the inner cylinder will be the same as that of any 
point on its axis. The differential of potential at any point 



62 



STATIC ELECTRICITY, 



on the axis P is that due to the charge on the circum- 
ferential differential of area of the small cylinder divided 
by its distance, minus the charge on the corresponding cir- 
cumferential differential of area of the large cylinder 
divided by its distance, or 

2 7r r qdz 2 ?r r qdz 
Vr*+z* " YE* + 2*' 
where z is linear dimension in the axial direction. It may 
be noted that the quantity on the elementary area of the 
larger cylinder is expressed in the same terms as that of 
the smaller cylinder. This is true since the bound charge 
of the larger cylinder is equal and opposite in sign to that 



dV 




Fig. 29. 



of the smaller, and it avoids evolving an expression for the 

surface density of the larger cylinder. The total potential 

at P is then (since z is measured in both directions) 

_ /•» t^itr qdz 2 izrqdz 

v - 2 y up 



+ z l YE 2 
z+ \/ r 



qdz\ 



a Ti z+ Yr i 4-z 2 ~X° 

4 7T r q\ log £ y ' " * I 

L ' z+Y2? 2 +zUo 



= kitrq log - — . 

e w 



CAPACITY. 63 

The quantity of electricity on the required portion of 
the inner cylinder is 

Q = 2rcrlq. 
Substituting, and introducing the dielectric constant 
n _ z. _Q _ i %*rlq _ , jj 

±kt q log — 2 log — 

A transmission line having considerable capacity re- 
quires a certain quantity of electricity to charge it which 
is not available for use at the far end. In direct current 
engineering this is of no consequence, since the line once 
charged remains charged while in operation. In alternat- 
ing current work the line is charged first positively them 
negatively, at each reversal of current the line discharg- 
ing in the direction from which it was last charged. In 
telephony the frequency of alternation is very high and the 
capacity of the telephone wires becomes a serious considera- 
tion, since a large part of the current that is to actuate a . 
distant instrument is consumed in charging the line while 
the potential is rising and is discharged when the potential 
is falling. A standard telephone cable containing two hun- 
dred pairs of No. 19 (B. & S.) wires, insulated with, 
dry paper and the whole surrounded by a lead sheath, 
has a capacity of about .085 microfarad per mile when 
measured between the two wires of a pair, and about .11 
microfarad when measured between one wire and all the 
others grounded on the sheath. Very recently Prof. Pupin 
has developed the relations which exist between capacity 
and inductance, and has shown where and how much in- 
ductance is necessary to balance the capacity of telephone 
lines. 



64. STATIC ELECTRICITY. 

In telegraphy little trouble has been experienced from 
capacity on ordinary land lines. In submarine work, how- 
ever, the capacity is enormous, being about one-third 
microfarad per mile. This multiplied by the transatlantic 
distance gives a capacity which effectually prevents the 
transmitting of impulses such as ordinarily are used in te- 
legraphy. Instead of the ordinary open and closed circuitl 
operation the cable is kept charged all the time and im- 
pulses are sent by very slightly increasing or decreasing 
the charge. Such transmission is necessarily slower than 
the ordinary method. 

34. Energy of a Charged Condenser. — The following 
reasoning can be employed to show the increase of energy 
in a system of conductors (a condenser for instance) when 
the charge in such system is increased. Suppose the quan- 
tity and the potential to be respectively Q\ and V 1 at the 
start, and Q % and V 2 at the end of the charging. 

We may consider the charging to be accomplished by 
bringing successive infinitesimal quantities of electricity, 
dQj up to the charged conductor from an infinite dis- 
tance. If the potential of the conductor at this particular 
time is V, then by the definition of potential, the work done 
in bringing this infinitesimal quntity up to it is 

d W - VdQ 
in which both V and Q are variables. At any instant the 
relation between Q and V can be expressed (from the 
definition of capacity). 

q= cv 

where C is the capacity of the condenser, hence 

dW=Vd[OV]. 
Summ ing up these infinitesimals of work from the original 



CAPACITY. 65 

conditions W t , Q lt T t , to the final conditions W 2 , Q 2 , V 2 , 
f W% dW C '= f v * VdV 

J Wx J Vx 

W 2 - Fi = i C(Vi -Y?) = iC(V 2 - FO (F a + FO 
But CV 1 =Q l and CV 2 = Q 2 , therefore 

= *(&+ eoc^.-'Fi)- 

If the condenser was uncharged at first, then TFi=0, 
(^ = 0, and V 1 = 0, and the energy of the system is 

35. Commercial Condensers. — In practice condensers 
are made in two classes, one for qualitative work and the 
other for quantitative experiments. There are four recog- 
nized dielectrics employed, air, paraffined paper, glass and 
mica. 

Air condensers are usually made up in some simple 
geometric form, such as concentric spheres or cylinders, 
whose dimensions can be readily found and their capaci- 
ties ascertained by calculation. They are sometimes called 
absolute condensers. Their capacities are small and they 
are used only in the laboratory. 

Standard mica condensers are the best for fine quantita- 
tive work. They are very permanent and accurate. In 
their construction alternate layers of tinfoil and mica are 
assembled so that every other sheet of tinfoil may be con- 
nected to form one conductor while those remaining form 
the other. This stack is then gently clamped together and 
left in a heated oven long enough to drive out all moisture. 
It is then quickly plunged into a hot bath of insulating 
material, after which it is left to dry in a vacuum chamber. 
5 



66" 



STATIC ELECTRICITY. 



This process insures the exclusion of all air in the dielec- 
tric. Condenser break-downs are frequently due to static 
electrification of enclosed air, the particles of which, bom- 
barding to and fro with the alternating charges, soften 
the insulation and lead to its final disruption. If air is in 
the dielectric, the condenser will get so hot after a few 
minutes' use on high voltages that the insulating compound 
will melt and run out. Mica is by far the best dielectric 
for moderate voltages, — say 250 volts alternating or 1000 




Fig. 30. 



volts direct, — extreme accuracy and permanence. It is 
also the most expensive. 

Fig. 30 shows a Leeds standard mica condenser, sub- 
divided into five sections of .05, .05, .02, .02 and .5 micro- 
farads respectively. This instrument is adjusted to an 
accuracy of J%, and is guaranteed to withstand 250 volts 
alternating indefinitely. The arrangement of the separate 
condensers is shown in diagram in Fig. 31. This arrange- 
ment permits of series as well as of parallel combinations, 
so that — as will be shown in section 37 — many values 
from .025 to 1 microfarad can be secured. 



CAPACITY. 



«7 



Paper condensers are much cheaper and hence can be 
built for much greater capacities. Save that paper is em- 
ployed instead of mica their manufacture is very similar 
to that just described. They may be made to withstand 
500 to 1000 volts alternating. In points of insulation, 
dielectric absorption and dielectric hysteresis they are in- 
ferior to the mica condensers. For these reasons they can- 
not well be used for standards. 

The most general use of condensers commercially is in 
connection with telephone circuits. For this purpose a cer- 
tain form of paper condenser has become standard. It 



.05 



h 



.05 



Fig. 31. 



consists of two very long strips of tinfoil about eight inches 
wide and two similar strips of paraffined paper about two 
inches wider than the tinfoil and somewhat longer. These 
are alternated and the four rolled up together in a flat form 
about 10" by 5" and of a thickness varying with the capac- 
ity, which latter is regulated by the length of the tinfoil 
strips. This form is treated with a baking process as be- 
fore described and is then inclosed in a metal case with an 
insulating top, on which are a pair of terminals respec- 
tively connected to the two pieces of tinfoil. Such con- 
densers are made in sizes from .5 to 5 microfarads and are 
capable of withstanding 500 volts direct. 



68 STATIC ELECTRICITY, 

High potential condensers have thick glass for the di- 
electric and frequently have copper foil instead of tin foil 
for the conductors. The thickness of the glass is necessi- 
tated by the great dielectric strength required and hence 
the capacity cannot be great. Such condensers are used 
experimentally in dealing with high tension, high fre- 
quency discharges from induction coils, static machines, 
and the interrupted spark from the secondary of a high 
tension step-up transformer. 

36. Electrolytic Condensers. — Two rods or plates of 
carbon immersed in a solution of zinc sulfate will act as a 
condenser of great capacity if not subjected to a higher 
potential than about one volt. Two such condensers similar 
in construction, connected in series, will work satisfactorily 
at double that pressure and so on. Two lead plates that 
are unformed, that is, not in the condition of storage bat- 
tery plates, immersed in dilute sulfuric acid, specific 
gravity about 1.2, will act as a condenser of great capacity 
if not charged at more than two volts. These electrolytic 
condensers have about the same volt-ampere capacity per 
unit volume as have the electrostatic condensers, but their 
use is limited because of the extremely low voltages at 
which they work. Like most electrolytic apparatus these 
condensers are sloppy and uncleanly, liable to cause dam- 
age by upsetting, and prone to unreliability of action. A 
particularly. convenient use to which they can be put is in 
reducing the sparking at the vibratory devices in induction 
coils. The action is this : At the moment of breaking, the 
circuit the self-induced e.m.f. which would tend to cause 
a spark is consumed in charging the condenser which is 
placed in shunt with the gap. When the circuit is re-estab- 
lished the electricity so stored is given back in useful form. 



CAPACITY. 



G9 



Another application is in transforming direct continuous 
current to direct intermittent current to secure greater 
heating effects. Suppose we have a surgeon's cautery to 
be heated. Arrange a set-up as in "Fig. 32. In series with 
a lamp between ordinary lighting mains is placed a con- 
denser. In shunt across the condenser is placed a circuit 
having the cautery and an ordinary vibrator in series. 
Suppose for a particular case the vibrator is adjusted so 
as to complete its circuit 1/10 of the time and to leave the 




Vibrator 



| Cautery 



Fig. 32. 



circuit open 9/10 of the time. If the resistance R f of the 
cautery is small in comparison with the resistance R of the 
lamp, — giving an approximately constant current circuit 
— and the condenser is of the proper capacity we will have 
the following condition : The current / will flow through 
the lamp continuously, but 9/10 of the time it is charging 
the condenser. Since the whole quantity of electricity must 
be transferred during the 1/10 part of the time that the 
shunt circuit is closed and the condenser is discharging, 



70 



STATIC ELECTRICITY. 



the current must be ten times as great as that in the lamp 
circuit, or 

T = 10 I. 
The heat developed in the cautery during the time of 
closed circuit is I' 2 R', or averaged through the whole time 

IW _ (10 1)*JZ >_ s 

io - W~ ~ 1U 1 M ' 

Thus we see that if the cautery were inserted directly in 
the lamp circuit, the power transformed to heat in the 



Fig. 33. 



cautery would be approximately only 1/10 as much as with 
the circuit shown. 

37. Connection of Condensers. — If two condensers are 
connected in parallel as in Fig. 33, it is evident that the 
combination is just equal to one larger condenser whosr 
capacity is the sum of the single capacities. 

If two condensers of capacities C 1 and C 2 are connects 
in parallel, their joint capacity C = C x X C 2 . 

If two condensers of capacities C x and C 2 are connected 
in series as in Fig. 34, a little consideration will show that 



CAPACITY. 



71 



when charged they must hold equal quantities of electricity, 
since any charge influenced on the right member of the left 
condenser must be balanced by an opposite and equal 
charge on the left member of the right condenser. But as 
they are equally charged, their differences of potential will 
be, by definition, inversely as their capacities. If Q be the 




Fig. 34. 

quantity of charge, and V the potential at which it is im- 
pressed, and C their joint capacity, we have: 

Q =V 1 C 1 = V 2 C 2 = VC (a) 

also V= V, + V 2 (b) 

Solving (a) 

and 



<h 



Substituting in (&) 



V t =YO 



r-Il + JS 
v c + ft 



or 



C = 



Ci C 2 



72 STATIC ELECTRICITY. 

If two condensers are connected in series, their joint 
capacity is equal to the reciprocal of the sum of the recipro- 
cals of their single capacities. 

The capacity of a series of condensers is always less than 
the capacity of any one of them, taken singly. 

It is interesting to note that the formula for capacities 
in series is analogous to the formula for resistances in 
parallel, while that for capacities in parallel is analogous 
to that for resistances in series. 

The capacity of a condenser of ordinary design can be 
approximately calculated by the following formula : — 

tf=. 000225 4^ 
t 

where 

C p= the capacity in microfarads, 

[ A = the area of dielectric between two conducting 

plates in square inches. 
n = the number of sheets of dielectric. 
t = the thickness of the dielectric in mils (1 mil 
= .001 inch), 
and h = the specific inductive capacity of the dielectric. 
The values of h for some of the more generally used dielec- 
trics are given on pages 53 and 54. 



CHAPTER IV. 

EXPERIMENTAL MEASUREMENT OF CAPACITY. 

38. Measurement of Capacities. — Whether the dielec- 
tric of a condenser is solid or fluid, the capacity is modified 
hy conduction and hy dielectric hysteresis, and is some 
function of the time of charging and the time between 
charging and measurement. Where these conditions are of 
any magnitude the capacity can be closely determined only 
by some method making use of rapid periodic charging and 
discharging. The direct conductivity of a condenser may 
be determined by the galvanometer as in ordinary insula- 
tion tests. Furthermore the capacity of the wires used in 
connecting up the condenser may be of such magnitude as 
compared to the capacity to be measured as to render it 
necessary to determine their capacity separately and sub- 
tract it from the observed capacity to give the true capacity 
of the condenser. 

In the electromagnetic system a capacity has the same 
dimension as the quotient of a time divided by a resistance, 
hence the suitable measurement of a time and a resistance 
.will determine a capacity. If the time be measured in 
seconds and the resistance in c. g. s. units the capacity will 
be expressed in c. g. s. units, while if the time and resist- 
ance be measured in seconds and ohms respectively, the 
capacity is given in farads. 

An application of this principle is shown in Fig. 35, 
where x and y are the coils of a differential galvanometer, 
and T a tuning-fork contact maker charging and discharg- 
ing the condenser ~k at a known frequency. The condenser 

73 



74 



STATIC ELECTRICITY. 



charges at the battery pressure, and discharges through the 
coil y. Coil x is constantly excited by a shunt current, 
Now, if the resistances ah and be are so chosen that the gal- 
vanometer does not deflect, and if we let 
B 1 — resistance of ab 



B t 



resistance of ac 



B g = resistance of coil x 



and n = number of vibrations per second of the tuning 
fork, then the capacity of the condenser K is 



C = 



nE,{E^li g y 




Ij^p 



K 



Fig. 35. 



The Wheatstone's bridge may be adapted to the measure- 
ment of capacity. In Fig. 36 one arm of a bridge is 
broken to make alternate contacts with a tuning-fork con- 
tact maker, which has the condenser to be measured in 
series with it, the arrangement being such that at one ex- 
tremity of the fork's vibration the condenser is charged 
while forming one arm of the bridge while at the other 
extremity this arm is opened and the condenser is dis- 



MEASUREMENTS. 



75 



charged through a short circuit, Letting a, h, c, g, and r 
represent respectively the resistances of their branches, and 
n the number of vibrations per second of the tuning fork, 
then the capacity of the condenser is 

a 2 



C = 



n 



be 



(a + c + g) (a + I + r) 



1 + 



ar 



c(a -H b + r. 



) \ 1 + b(a + c + g)) 




Fig. 36. 



If ~b, c, and # are about equal, the formula reduces ap- 
proximately to 

a 1 



C 



n b c 



1 + 



ag 



b(c + g) 

Further, if b and c are taken very large (10 5 to 10 T 
ohms), a and r relatively very small (10 to 10 3 ohms), and 



76 



STATIC ELECTRICITY. 



g intermediate between them, (say 10 4 ohms), then ap- 
proximately 



C = 



nbc' 



As in measuring resistances in the Wheatstone's bridge, 
the battery connections should be reversed between dif- 




Fig. 37. 



F. 



ferent observations to avoid error due to thermal e. m 
generated in the bridge arms. 

Another bridge method which determines simultaneously 
two capacities is shown in Fig. 37. Shunted around arm 
c is a condenser C l9 while in series d is a condenser C 2 . 
Arms a and b are in a bridge wire on which plays a sliding 
contact going to what is usually the galvanometer circuit. 
The " optical telephone " is however used, since it is to 
detect alternating currents. This is a telephone upon 



MEASUREMENTS. 77 

whose diaphragm is affixed a small mirror which reflects 
an illuminated slit upon a screen. The image will, of 
course, move violently for very small displacements of the 
diaphragm. An alternating e. m. f. is applied to the 
terminals of the wire a h, and the resistance c and the slid- 
ing contact are manipulated until no current flows in the 
telephone. If a, b, c, and d represent respectively the re- 
sistances of their branches, and / equals the frequency of 
the alternating e. m. f. (found by multiplying the num- 
ber of revolutions per second of the generator by half the 
number of its poles), then 



01 = 


1 

_ i 


V 


h - 


— a d 
ad ' 


a 




a 



j~ d (b g — a d) 
39. Comparison of Capacities. — In determining a ca- 
pacity it is usual to compare it with a known capacity, this 
being as a rule quicker and more simple than measuring its 
absolute capacity. 

The simplest application of this principle requires 
merely a cell of constant e. m. f. and a ballistic galvanom- 
eter whose constants need not be known, a known capacity 
C\, and the capacity to be compared C x . The procedure 
is as follows : Charge C t from the source of constant 
e. m. f. and discharge through the galvanometer, noting 
the deflection t . When the galvanometer is again at rest, 
repeat with C x , noting the deflection ° x Then 
C x __ sin \Q X 
C x ~~ sin $ e x ' 
This method is quick and fairly accurate if a good con- 
denser is used as C v The most accurate results are ol> 



78 



STATIC ELECTRICITY. 



tained when the condensers are of almost the same 
capacity. 

Fig. 38 shows the method of Cohn and Arons. A 
quadrant electrometer connected " in quadrant " is used, 
i. e., the very small pressure to be observed is used to charge 
one diagonal pair of quadrants, while the other pair is kept 
at zero potential by an earth connection. The needle is 
charged to a high potential either by a many cell battery 
or a highly charged Ley den jar, the remaining side of the 
battery or jar being connected to earth. This is the he- 
mostatic method of operation described in section 43. This 
instrument must be calibrated so that the potential at which 




the quadrants are charged is known for any given deflec- 
tion. In the set-up a cell of known e. m. e. has one pole 
connected to the earthed pair of quadrants and the other 
pole to the condenser ; the remaining pair of quadrants is 
connected to the other side of the condenser through a key 
h z . A third branch containing a key hi shunts the electrom- 
eter and k 2 . The two keys are designed to be opened in 
quick but measurable succession, and this is accomplished 
by a pendulum. Now if 

E = e. m. f. of the cell in volts, 

V = potential of quadrants after opening the keys, 



MEASUREMENTS. 79 

t — time in seconds between opening lc x and h 2> 
R c = olimic resistance of the condenser, 
c = capacity of one pair of quadrants in farads, 
and C =s capacity of the condenser, in farads, then 

(C+c)fi e = t - F ~. 

log iTT 
If c and R are known this is an absolute measure of 
(7, but if they are not, the experiment is repeated twice, 
first with a condenser of infinite ohmic resistance (an air 
condenser whose capacity G' can be determined by linear 
measurement) placed in parallel with 0, and second with 
a large known resistance R placed in parallel with the 
original condenser. From the second and third observa- 
tions we have 

(C+ C + c)£ c = ^gr- 

and (G + c) {E + B) = 



log 



E-V 



G c 
From these three equations values for— 7 , 777, and fi c 

may be obtained. This method is free from errors due to 
conductivity and dielectric hysteresis in the condenser. 

De Sauty's bridge method is shown in Fig. 39. Two 
condensers occupy two arms of the bridge and two resist- 
ances — which must be high to attain accuracy — are in 
the other two. Any delicate galvanometer may be used. 
A switch is arranged so that c may be connected to h, thus 
charging the condensers, or to a thus discharging them. 
When the resistances are so adjusted that no deflection 



80 



STATIC ELECTRICITY. 



occurs at charge and discharge then 

This method is independent of the battery voltage, in fact 
a source of alternating e. m. f. may be substituted for the 
battery and switch if the galvanometer is replaced by a tele- 
phone, an electrometer or an eleetrodynamometer. 

Thomson's method is somewhat better. As shown in 
Fig. 40 the battery circuit is uninterrupted and a key is 
opened and closed in the galvanometer circuit. When this 
causes no movement of the needle, then 

If one of the condensers, C v has a material conductivity 




Fig. 39. 



its capacity can still be measured by using the optical tele- 
phone in place of the galvanometer and key and applying 
a sinusoidal e. m. f. instead of a battery e. m. f. A 
resistance R 9 must also be placed in series with the other 
condenser C 2 . Then if B is the ohmic resistance of the 



MEASUREMENTS. 



81 



condenser C and / is the frequency of the impressed 
e. m. f. then 

and the ratio of the capacities 

74 = T£ ~ i^ " -r; -J ^ Cl C » 

The last member of the right side is a correction member 
for which an approximate value for C l C 2 is sufficient. 

This method as well as others using an alternating 
charge and discharge of high frequency is apt to give 




+!#M'tf 

Fig. 40. 

erroneous results when applied to condensers which have a 
considerable charging period, that is, require considerable 
time to receive their full charge after the e. m. f. ha3 
been impressed. Cables, particularly submarine, have a 
very considerable charging period. 

To determine if two condensers are of the same capacity 

by means of alternating currents and the telephone, the 

comparison resistances of the de Sauty method may be 

made infinite, reducing the set-up to the simplicity of Fig. 

6 



82 



STATIC ELECTRICITY. 



41. If the capacities are equal the diaphragm of the tele- 
phone will not move. This method is very rapid and quite 
accurate for testing a number of capacities of supposedly 
the same value. One of the capacities may be made vari- 
able and calibrated and the others all balanced up against 
this. Another arrangement is shown in Fig. 42. 




The comparison of capacities by separation of the charge 
was the method used by Farady in his original investiga- 
tions. Let Q represent the quantity of electricity held by 
a condenser of C t capacity when charged to a potential E. 
ISTow disconnect the condenser from the source of e. m. f. 




Fig. 42. 



and connect it to a second condenser which was previously 
unchanged. If q represent the quantity now held by the 
first condenser, and C 2 is the capacity of the second, then 

A_ i 

G % ~ Q-q 

If C, is very small as compared with C v C l may be dis- 



MEASUREMENTS. 83 

charged into C 2 n times and C 2 entirely discharged after 
each of its n charges. Then if q is the quantity in C 2i after 
the nth discharge, 

n 

— ^y— — n n 

6 * v~q— vy 

This method may be used to compare the capacities of 
cables with normal (air) condensers, but is liable to great 
errors due to leakage unless the successive discharges are 
made very quickly. 

The capacity of any condenser can be directly measured 
with a good degree of accuracy by means of a suitable 
calibrated ballistic galvanometer as described at the end 
of section 41. 



CHAPTER V. 

INSTRUMENTS USED IN ELEOTItOSTATCS, 

40. Ballistic Galvanometer. — An instrument for detect- 
ing and measuring electric currents or quantities by virtue 
of the electromagnetic reactions of such currents is called 
a galvanometer. Of these there are two general classes, 
those in which the current in a fixed coil deflects a pivoted 
magnet, — as for instance, a compass needle, — and those 
in which the current in a coil suspended in a fixed mag- 
netic field causes the said coil to move. An instrument of 
this nature having no permanent magnet, but consisting of 
a fixed coil and a suspended coil, is generally called an 
electro dynamometer. Electrostatics play no> part in the 
operation of this instrument, but it is extensively used in 
the observation of electrostatic phenomena. 

A galvanometer whose moment of inertia is so great that 
the entire transitory current due to passing a charge 
through it may be considered to have passed before the 
needle has moved sensibly from its position of equilibrium 
is called a ballistic galvanometer. When such an instru- 
ment is used, not the steady deflection is read, but the ex- 
tent of the first throw or hick, and as the motion is slow, 
the extent of such throw is easily observed. In the early 
types of galvanometers where the earth's magnetism was 
the controlling force, a needle was made ballistic by weight- 
ing it, or by making it in the form of a sphere. In the 
modern D'Arsonval or swinging coil type, slow motion is 
secured by making the coil sufficiently heavy. 

Like the theoretical pendulum, an unimpeded galvanom- 

84 



INSTRUMENTS. 85 

eter needle or coil would oscillate forever, once it had 
been started. Practically there are several causes which 
tend to bring the needle to rest, among which are; — the 
friction of the air, the friction of the pivot or the only 
partial elasticity of the suspending fibre as the case may 
be, and most of all, the generation of electric currents in 
adjacent conductors due to the changing magnetic field. 
This last influence is most clearly shown in the case of a 
suspended coil galvanometer. If this coil be moving in the 
permanent magnetic field of the instrument, it will act like 
the armature of a dynamo, and have electromotive force 
set up in it, due to its conductors cutting magnetic lines of 
force. If the coil be on open circuit, no current can flow, 
and the retarding or damping effect will be small. If how- 
ever the coil be connected with other apparatus so that it 
forms part of an electric circuit, then current will flow due 
to this electromotive force, such current doing work in 
heating the wires and absorbing energy from the moving 
coil till the latter is brought to rest. The extent of the 
damping will depend on the resistance of the circuit; the 
less the resistance, the greater the current and the heavier 
the damping. Galvanometers of the swinging magnet type 
can be similarly damped by bringing masses of copper in 
the vicinity of the moving needle in which Foucault or 
eddy currents may be generated, thus wasting the energy. 
Galvanometers for less delicate work may be damped by 
having a projection from the moving member equipped 
with fan-like vanes or dipping into mercury or some more 
or less viscous oil. The method of damping by supplying 
a suitable outside circuit is the most usual with modern 
instruments. To bring the coil to rest preparatory to tak- 
ing a reading, it is customary to short-circuit the wires 



86 STATIC ELECTRICITY. 

leading to the galvanometer, thus giving the heaviest pos- 
sible damping. 

41. Calibration of Ballistic Galvanometer. — To calibrate 
a ballistic galvanometer is to determine an expression in- 
volving Q, the quantity of electricity passed in a transi- 
tory current through the galvanometer, and fi the throw of 
the needle due to this quantity, such that the quantity 
passed can be determined for any observed throw. 

As shown in Gray's " Electricity and Magnetism " or 
in Maxwell's " Electricity and Magnetism/' from a con- 
sideration of the equations of mechanics relative to the 
motion in bodies due to various forces, the equation for a 
damped galvanometer of the magnetized needle type can 
be written 

X n 

7-7/77 — arc tan — 

n _ JOL 1 . /? £ TZ X 

q -~a v^+ir sm — » 

in which 

H — the horizontal intensity of the magnetic field in 
which the needle swings, 

G = the moment of the forces with which a unit current 
in the coils acts upon the needle when both coils and needle 
are in the same vertical plane, 

T— the time of one complete oscillation, as for instance, 
from one maximum elongation to the next maximum 
elongation in the same direction. 

X — the logarithmic decrement. The amplitudes of the 
needle's oscillations are in a diminishing geometrical 
progression. \ is the natural logarithm of the common 
ratio of this progression. Thus if one amplitude (to the 
right) be a t and the next (to the left) be a % then X = 



log* 



a* 



INSTRUMENTS. 87 

The only arbitrary assumption included in the above 
formula is that the damping is proportional to the velocity 
of the swing of the needle. 

If the deflection of the needle is small, then the sine of 
the angle can be taken as equal to the angle, and the above 
formula becomes, 

n _ II T — arc tan— 

If the logarithmic decrement is small then tan - 1 — = 
tan -1 oo — r nearly 
and 

X _ 1 TZ X 

n Un T T XX' P 

= • = 1 -+a + 4 + 48 + •••• 
but as X is small the higher powers may be omitted and 
the equation last written reduces to 

No sensible error is introduced by using this last equation 

if A < i and p < 30° or J. 

b 

If there is no damping at all, x —0 and the original 

equation reduces to 

n H T . fi II T 

if the deflection is small. 

To calibrate a ballistic galvanometer by means of the 

above equations it is necessary to know -p , T and X . The 



88 STATIC ELECTRICITY, 

TT 

value of — - y may be determined by passing a uniform cur- 
rent through the galvanometer and observing the steady 
deflection. Assuming, as is the case with most galvanom- 
eters, that the current is proportional to the tangent of 
the angle of deflection 

i — -— tan 8 

G 

where d is the deflection and i the current in c. g. s. units. 
If the current be measured in amperes, I, then 
H_ _ 10 7 
G tan d 
The current I will be very small and can best be measured 
by the fall of potential across some large and known resist- 
ance in the circuit, perhaps the resistance of the galvanom- 
eter itself, but better if possible, some much larger resist- 
ance in series therewith. The fall of potential must be 
measured by some delicate means, such as balancing 
against a Clark standard cell on a potentiometer. 

T, the period of oscillation, can be determined with any 
degree of accuracy by taking the time of a sufficient num- 
ber of oscillations and dividing this time by the number of 
oscillations. These are complete vibrations, that is, from 
one extremity of the swing to the other and back again to 
the first. 

The logarithmic decrement, X, is not a constant in any 
one galvanometer, but is some function of the circuit in 
which the galvanometer is included. For, as explained in 
a previous paragraph, the amount of damping depends 
upon resistance of the circuit, being approximately in- 
versely thereas. As suggested above, two consecutive 
amplitudes could be observed and the natural logarithm of 



INSTRUMENTS. 89 

the ratio of the first to the second would give the value of 
A.. More accuracy however can be obtained by reading a 
number of successive elongations. These will form the 
terms of a decreasing geometrical series, and if a be the 
rath term, and a n be the nth term the natural logarithm of 
the common ratio of the series is 

log £ ratio — X = (l°ge a m — l°ge a n)- 

n — m 

By letting the different observed values be in turn a m and 
a n , from a comparatively few observations a number of 
values of x may be determined, whose average may be 
assumed as the correct value of x . This process should be 
repeated with different resistances in the circuit, and a 
curve plotted with values of R and A respectively as ordi- 
nates and abscissas. Then in any experiment, with a knowl- 
edge of the resistance in circuit the value of I can be de- 
termined without further calculation. 

It must be remembered that in all the foregoing by " de- 
flection " is meant the angular displacement of the needle 
from its position of equilibrium. If a mirror galvanom- 
eter, throwing a spot of light on a straight scale is used, 
then the scale reading is proportional to the tangent of 
twice the angle of deflection. 

A galvanometer may also be calibrated by the inductor 
method, which is as follows: If a coil of wire of n turns 
be rotated in a magnetic field so as to cut lines of force, 
an e. m. f. will be induced therein whose value in volts 
(see Sheldon's Dynamo Electric Machinery, § 13), is 

T? - JUL c ll 
io 8 dt 

where -~ is the rate of cutting of lines of force. If such 

Out 



90 STATIC ELECTRICITY, 

a coil be part of a circuit including the galvanometer with 
a total resistance of B ohms, then the current 
T _ E nd<p 



B ~WBdt 
from which 



Idt 



nd(p 



10 8 Y2 
Integrating between the limits / and f V d<P, 

It = Q = 7-75-7^ coulombs. 

? is the number of lines of force cut by the coil which 
are active in producing e. m. f. If any line of force is 
cut once by a part of the coil, and is then cut a second time 
by the same part (not the opposite side), the e. m. f. of 
the second cut opposes that of the first, and such a line 
must not be counted as active in producing e. m. f. If 
the inductor be a coil of n turns inclosing an area of A 
square centimeters, if it be placed perpendicular to the 
direction of the horizontal component H of the earth's 
magnetic field, and if on being revolved 180° in that field 
it produces a kick a in the galvanometer then 

<P = II A 
and 

n _ 2nRA 

* " 10 8 ^ " 
(As each turn cuts each line twice, once by its upper half, 
and once by its lower half, the number of conductors cut- 
ting lines is twice the number of turns.) But from the 
previously written approximate equation of the galvanom- 
eter 






INSTRUMENTS. 91 

Then if K is the constant of the galvanometer 
II T 2n HA 



K = 



2 * G 10 8 i?(l + ^)a 



The formula for this galvanometer can now be written 
simply 

and if the instrument is not damped, 

Q = KP. 

The method just outlined is called the " earth inductor " 
method. The chief objection to it is that the value of H is 
difficult to determine, besides which it is not a constant as 
to time or place. The " current inductor " method avoids 
this difficulty. Let a long solenoid be formed of closely 
wound wire, there being n t turns per centimeter length of 
solenoid. If this is placed east and west and a current of 
I amperes be passed through it, the number of magnetic 
lines of force per square centimeter cross section set up 
therein is 

4 * U X I 

10 
and if the mean area of the solenoid be A, the total flux 

4:7V n, 1 A 

<p = 1 

10 

If a small test coil of n 2 turns be wound closely around the 
solenoid and with the galvanometer forms a circuit of R 
ohms total resistance, then the quantity of electricity passed 
through this circuit on starting or stopping the current I t 
in the solenoid is (see Sheldon's Dynamo Electric Ma- 
chinery, §§ 13, 16.) 

n _ n 2 <p _ 4 7T Tix n 2 Ii A 

V " 10 8 i? " 10 9 i? 



92 STATIC ELECTRICITY. 

from which, as before, the constant of the galvanometer, 



K= 



2 7T U- 



10 9 R (1 + 



where a is the kick corresponding to the current I x in the 
solenoid. 

This method is especially convenient for calibrating 
D'Arsonval galvanometers, which do not necessarily follow 
the ordinary ballistic formula. Here various currents may 
be passed through the solenoid and the quantities corre- 
sponding to the various kicks determined. A curve can then 
be plotted between the two which can be used subsequently 



SL. 



<S> 



— Rheostat 



Fig. 43. 



for all observations where the external circuit has the same 
resistance as when the calibration was made. For other 
resistances other curves must be drawn. 

Still another method of calibrating a ballistic galvanom- 
eter is by means of the charge of a standard condenser. 

To accomplish this an arrangement of apparatus such as 
is shown in Fig. 43 is employed, the particular feature of 
which is the quadruple contact key. This key is normally 



INSTRUMENTS. 93 

held up against a contact. In this position the galvanom- 
eter circuit is open and the condenser is in series with a 
charging battery of constant potential. As the key is 
pressed down three things occur: First, the battery circuit 
is broken ; second, the condenser is discharged through the 
galvanometer ; and third, the galvanometer circuit is closed 
through an appropriate amount of resistance in a rheostat 
to form the damping circuit. 

Now the potential, E, to which the condenser was 
charged may be determined ; the condenser capacity, C, is 
known ; and the kick, a, is observed. 

Then Q = EC=Ka 

.-.IT-** 

a 

where K is a constant directly reducing galvanometer de- 
flections to quantity units according to the formula Q = 
JTj3. This assumes what is practically true of modern 
D'Arsonval galvanometers, that the throws are proportional 
to the quantities causing them where the resistance of the 
damping circuit is the same in each case. 

42. Kelvin's Absolute Electrometer. — Two discs oppo- 
sitely charged will attract each other according to a definite 
law, and hence can be made to measure the difference of 
potential between them. The attractive force exerted by a 
charged point varies inversely as the square of the distance, 
because the lines of force emanate radially, while from an 
indefinitely extended plane the lines are parallel and the 
force is independent of the distance (§ 16). In Kelvin's 
attracted-disc electrometer there are two discs and a ring, 
arranged as in Fig. 44. B is a balanced disc closely sur- 
rounded by the fixed guard-ring G. These two are charged 
to the same potential and are practically one plate. The 



94 



STATIC ELECTRICITY. 



lines of force may be considered to emanate perpendicu- 
larly from all parts of this plate save those parts near the 
edge, where the lines become divergent. Hence it may be 
assumed that the lines emanating from B are all parallel 
and the force of attraction between B and M is independ- 
ent of the distance between them when this distance is 
small in comparison with the diameter of B. 

M is a disc that may be moved up or down by means of 
a micrometer screw. Now when B and G are at one po- 
tential and M at some other there will be an attraction 
which causes B to descend. This force could be measured 



G 



A 

Fulcrum 



O 



1 

M. 




— 


1 =3 








i 






— 




m\K- . 



Fig. 44. 



by counter-balancing B but better by moving M away till 
B returns to its normal plane. This does not negate the 
statement previously made that for parallel lines the force 
is independent of the distance, because moving M changes 
the electrostatic capacity of the combination, hence changes 
the quantity of charge for a given potential, which in turn 
changes the number of lines of force or the attractive force. 
In the practical forms of these instruments, one of which 
is shown in Fig. 45, many refinements are introduced, chief 
of which is an arrangement of lenses by which the level of 



INSTRUMENTS. 95 

B can be accurately determined. The absolute electrometer 
differs from the attracted-disc electrometer only in that the 
counterweight is replaced by an arrangement of springs, 
whereby the attractive force may be measured in dynes. 




Fig. 45. 

Lord Kelvin describes two methods of using the instru- 
ment. The idiostatic method consists in keeping M and B 
at the potential difference to be measured (G being at the 
same potential as B in every case) and adjusting M till B 
maintains its proper level. 



96 STATIC ELECTRICITY. 

The difference of potential, V 1 — V 2 is, according to sec- 
tion 24, 

V 1 - V 2 = ±n<rd. 

If S be the area of the movable disc, B, 8<r is the quantity 
of electricity on it, and again from section 24, it will be 
attracted to the fixed plate with a force 

F = 2 7t a X 8 * 
From which 

~~ ~F~ 



/ 



2*8 
Substituting this in the above equation, 

F, - V % = <V^ 

If the values of <#, F, and 8 are expressed in centimeters, 
dynes, and square centimeters respectively, then the po- 
tential difference will be expressed in absolute electrostatic 
units, which multiplied by about 300 will become volts. 
(See Appendix.) 

The absolute distance, d, is hard to determine, hence the 
Jietrostatic method is employed. G and B are kept con- 
stantly charged to a certain potential, and M is connected 
successively with the two points whose potential difference 
is to be determined. If F j and V 2 be the potentials of the 
two points, and d t and d 2 the respective distances which 
bring B to a balance, then the required potential difference 

7i - f, = % - d % ) y^S- 

dj— d 2 being merely the difference in micrometer readings 
it is easily obtained. 

43. Quadrant Electrometers. — The principle of Kelvin's 
quadrant electrometer is shown in Fig. 46. Four brass 



INSTRUMENTS. 



97 



quadrants are arranged as shown in a horizontal plane. 
Small air gaps are left between them and the alternate 
quadrants are in electrical connection. Just above the 
quadrants is a centrally suspended vane of thin alumi- 
num, shaped as shown. Although it is neither magnetic 
nor needle-shaped this vane is frequently called the needle 
of the instrument, because of its homology to the old type 
galvanometer needle. When used idiostatically, one pair 




Fig. 46. 



of quadrants and the vane are kept at one potential and the 
other pair at some other potential, usually that of the earth. 
The first pair will repel the vane and the second attract 
it, owing to the arrangement of the charges, and the angu- 
lar displacement may be made a measure of the difference 
of potential. This arrangement will only measure poten- 
tial differences of 30 volts or more. The hetrostatic method 
is much more delicate and is more frequently used. If 
T^l — T 7 ^ is the difference to be measured, one pair of quad- 
Sri 



98 STATIC ELECTRICITY. 

rants is charged to Y h the other pair to Y v and the vane to 
Y$, V z being much greater than Y x or V 2 . If now c be 
the capacity per unit angle of the condenser formed by a 
portion of the vane well within the qnadrant and away 
from the vane's edges, and the vane's deflection is in- 
creased by de, so that an angular portion do passes from 
each quadrant of one pair to the quadrants of the other 
pair; then the corresponding increment of energy, by 
section 34, is 

d W = i c do ( V 3 - V 2 f - i c do ( V 3 - V 1 ) 3 

or dW ,-rr tt-v / T7 . Vi+ Yi 



de 



0(^-^(7-3--^). 



"Now , is the moment of the force producing the deflec- 
tion e and must be equal to the moment of torsion resist- 
ing the deflection. The moment of torsion in either mono- 
or bifilar suspension may be considered proportional to the 
angle, for small angles, hence 

dW 

-do = c ^' 
Furthermore V 5 is usually so great that the expression 

Y B - ? may be considered constant, so we can let 

Y Vl+Vi -c, 
Vs 2 - c* 

Substituting these values, we have 

c l 6 = c(Y 1 - Y 2 )c 3 

or, letting — = m 

Y,- Y z = me % 



INSTRUMENTS. 



99 



m is tlie constant of the instrument and may be deter- 
mined by using a known difference of potential for 
V 1 ~~ V 2 and observing the corresponding 6. 

In the practical instrument many refinements are in- 
troduced. Bifilar suspension is used. Two sets of quad- 
rants are arranged one above, the other below the vane ? 




Fig. 47. 



the peripheral edges of the corresponding members of the 
upper and lower set being united, forming a sort of cylin- 
drical box around the vane. Readings are taken by a beam 
of light reflected along a scale by a vertical mirror attached 
to the vane. The whole is mounted in a glass vessel the 
outer side of which is coated part way up with tinfoil. 
Concentrated sulfuric acid is inside and the combination 

LofC. 



100 STATIC ELECTRICITY. 

forms a Ley den jar, which may be highly charged to a 
potential V s and then made to keep the vane at this poten- 
tial. The sulfuric acid also serves to keep the atmosphere 
within the jar very dry. An auxiliary attracted-disc elec- 
trometer is mounted on the apparatus and is called the 
gauge. It serves to indicate the potential of the Leyden 
jar V 3 . A circular level and levelling screws are provided. 
Fig. 47 shows Kelvin's quadrant electrometer in its com- 
mercial form. 

Prof. H. J. Ryan has designed a quadrant electrometer 
of unique form, shown in Fig. 48. The quadrants are set 
in quartz discs, which in turn are mounted on the two cir- 
cular metal cover plates, and are faced with mica to pre- 
vent short circuiting the instrument should the vane get 
out of place and touch them. The cover plates are held in 
position by spring clips, three pegs being provided over 
which each cover fits, insuring that the quadrants always 
maintain the same positions. Opposite quadrants are 
automatically connected on assembling the instrument by 
metallic strips which make contact with binding posts 
mounted on the quartz discs and appearing on the outside 
of the cover plates. The " needle " in this case is an alu- 
minum cylinder with quadrangular portions cut away. A 
mirror is mounted upon it so that its silvered surface lies 
in the axis of rotation. The cylinder may be suspended by 
a metallic fiber and charged through it, or by silk or quartz 
fiber, in which case the connection is made by means of a 
very fine platinum wire, running loosely from the cylinder 
to a hook, which latter is connected with an external in- 
sulated binding post. The method of making this fine 
wire is of interest. A very fine platinum wire is heavily 
silver-plated. The wire thus formed is drawn out as fine 



INSTRUMENTS. 101 

as might be and the silver then dissolved off by suitable 
acids. The insoluble platinum core is then found to be 
a wire only .0002 inch (.2 mil) in diameter! Rigidly 
connected to the aluminum cylinder, and exactly in the 



Fig. 48. 



center of the containing case is a small magnetic needle. 
On the outside of the case and symmetrically disposed 
with reference to the needle are two coils of insulated wire 
of different numbers of turns, whose terminals are insul- 



102 STATIC ELECTRICITY. 

ated binding posts. The instrument may be used in two 
ways: either the e. m. f. or potential difference may be 
calculated by a deflection method as in the ordinary quad- 
rant electrometer; or a zero method may be employed, a 
measured current being passed through either or both of 
the coils, acting on the needle and bringing the system to 
zero. Thus the voltage can be measured by a properly 
calibrated ammeter in the restoring circuit. By this 
method potentials as high as 5,000 volts may be measured. 
Prof. H. S. Carhart has devised a variation of the quad- 
rant electrometer which uses a zero method, the returning 
force being supplied by applying torsion to the suspension, 
exactly as in the electrodynamometer. The cylindrical 
box-like set of quadrants of the Kelvin instrument is re- 
placed by two similar boxes mounted coaxally and con- 
sisting of semicircles instead of quadrants. The two vanes 
are of aluminum and are also semicircular. They are 
mounted diametrically opposite on a rigid aluminum wire, 
and at such a distance apart that one may play freely in 
each box. The suspension is a phosphor-bronze wire 
.0015 inch in diameter, terminating in a hard rubber tor- 
sion head. A pointer attached thereto plays over a gradu- 
ated, completely circular scale. A platinum-silver wire 
spiral connects the lower part of the " needle " to a pin at 
the bottom of a cup of paraffin oil. The oil acts as a 
damper and the whole coil serves to conduct the charge to 
the " needle." A mirror is mounted on the wire connect- 
ing the two semicircular vanes. The opposite portions of 
the two boxes are cross-connected and attached to external 
binding posts. The vanes are also connected to one of the 
binding posts, compelling the use of the instrument idio- 
statically. When the instrument is charged the " needle " 



INSTRUMENTS. 



103 



deflects, is brought back to zero by turning the torsion head, 
and the angle turned through is read from the scale. A 
calibration curve will give the impressed pressure in volts. 
Aryton and Mather's zero electrometer is similar to 
the one last described, in so far as the returning force is 
concerned, but differs in that the deflecting force is due to 
attraction, not repulsion. The " needle " is a rectangle of 
aluminum ribbon, through the centers of the short sides 
of which is passed a rigid axis of the same material. The 
axis is suspended from above by a phosphor-bronze fibre 
terminating in a torsion head with index and scale. To 
the lower end of the axis a light pointer is attached at 




*s* 



Fig. 49. 



right angles which plays over a zero mark. Parallax is 
avoided by placing a mirror under the needle, enabling 
the observer to determine when he is normally above it. 
Around each vertical side of the " needle " is bent a strip 
of metal termed an inductor. An axially normal section 
of the " needle " and inductors would appear as in Fig. 
49. The inductors are connected to one binding post and 
the " needle " to another. Any difference of potential set 
up between them will give the " needle " a tendency to 
swing further into the inductors. This is countered by 
turning the torsion head till the pointer is on zero, when 



104 STATIC ELECTRICITY. 

the scale at the head can be read and the required value 
determined from a calibration curve. 

All the instruments described above have the advantage 
that they require no current, wherein they differ from 
galvanometers. Hence they are adapted to electrostatic 
measurements where, as a rule, no current is to be had, for 
electricity in motion cannot be part of a static phenome- 
non. Furthermore their capacities are so small as not to 
materially reduce the charge of any moderately charged 
body, and for the same reason, when used for alternating 
current measurements, they take almost no charging cur- 
rent. On direct currents they take absolutely no power. 
Since their indications are independent of any electrical 
resistance within them they are independent of tempera- 
ture variations. 

44. Capillary Electrometers — The principle of the cap- 
illary electrometer was first stated by Lipmann, but the 
instrument has never attained the wide popularity of those 
previously described. This principle is the change of sur- 
face tension between mercury and an electrolyte (dilute 
sulfuric acid generally) when the difference of potential 
at the point of contact changes. The instrument consists 
of two quantities of mercury electrically connected by 
sulfuric acid, the contact surface at one quantity being 
comparatively large, that at the other very small, in a 
capillary tube in fact. If now the smaller surface as- 
sumes a position of equilibrium due to the different heads, 
capillarity, and surface tension, then a difference of po- 
tential between the two liquids will cause the meniscus to 
move to some other position. The effect of a potential 
difference between the two quantities of mercury will 
practically all be felt at the smaller surface. 



INSTRUMENTS. 



105 



A simple laboratory form of instrument is shown in 
Fig. 50, which is practically self-explanatory. Under the 
capillary tube T is a finely divided scale, which with the 
aid of a lens allows small deviations of the meniscus to 
be noted. The whole is mounted on an inclined adjust- 
able board, so that the position of equlibrium can be regu- 
lated more easily than by removing or adding mercury to 
A. The wire that leads to B must be insulated from the 
sulfuric acid, preferably by sealing in a glass tube. The 
capillary is about .5 mm. bore, and the acid is diluted 1 




Fig. 50. 



to 6 by volume. The delicacy of the electrometer is less 
the more inclined the capillary is. When horizontal an 
accuracy of .001 volt may be attained. The faults of this 
instrument will be discussed later. 

A much more practical type is shown in Fig. 51. This 
is as nearly portable as an instrument comprising three 
liquids and glass parts can be. T is a thick test tube hav- 
ing a platinum terminal fused in its lower end. This 
terminal is covered with a layer of mercury and the rest 
of the tube is nearly filled with dilute sulfuric acid. The 



106 



STATIC ELECTRICITY. 



top of T is closed by a tight-fitting rubber cork, through 
which projects the capillary tube C, made by drawing out 
a thermometer tube. This must rest close against the side 
of T to avoid troublesome refractive effects. A piece of 
flexible rubber tube B is fitted over the top of C and 
wrapped to prevent leakage. This is spanned by a screw 




Fig. 51. 

pinch-cock P and is filled with pure mercury. Its upper 
end is closed by a piece of glass into which is fused the 
second platinum terminal. On compressing P the mercury 
is forced through C, and on slightly releasing the pressure 
it draws part way up, bringing up the acid after it. The 
whole is now ready for use. The microspope M fitted 
with a micrometer eye-piece may now be adjusted to bring 



INSTRUMENTS. 107 

the meniscus to the hair line. The key K serves to keep 
the instrument short-circuited save when in actual use, 
a very necessary condition. Mr. C. F. Burgess, who de- 
scribed this instrument before the American Institute of 
Electrical Engineers, claims for it a sensible deflection for 
.001 volt and suggests the possibility of ten times that 
sensitiveness when certain refinements are observed. 

This instrument cannot be used quantitatively, because : 
equality of bore is impracticable, the deflection is not 
necessarily proportional to the potential difference, it is 
impracticable to maintain always the same density of acid, 
and most important of all, the instrument cannot be used 
on pressures exceeding one volt, for then electrolysis sets 
in and the readings become worthless. The mercury in 
the capillary should suffer cathodic polarisation, i. e., be 
connected to the negative side, if any attempt be made to 
use it quantitatively. Cathodic polarisation of over one 
volt causes the formation of hydrogen bubbles, while ano- 
dic polarisation of over a few hundredths of a volt causes 
the mercury to become - impure and the meniscus to lose 
its mobility. 

All the facts indicate a usefulness for the capillary elec- 
trometer only as a zero instrument in some compensation 
method. While its accuracy is not so great as that of the 
finest galvanometers, it may be used where they cannot 
because it is only slightly sensitive to mechanical, almost 
indifferent to electric, and completely indifferent to mag- 
netic disturbances. Furthermore, it is practically dead 
beat. 

It is inevitable that the capillary electrometer will be 
overpolarised by accident when in operation, and then 
the one last described has a great advantage over the 



103 



STATIC ELECTRICITY. 



simple form first described. In the first arrangement it- 
would cost some pains to remove a bubble or drop of mer- 
cury from the capillary, while in the second instrument 




Fig. 52. 



it is merely necessary to compress the pinch-cock till a 
drop of mercury falls from the capillary into the bottom 
of T when the instrument is in its original condition. 



INSTRUMENTS. 



109 



45. Electrostatic Voltmeters — 'Any non-zero electrom- 
eter may be calibrated to read directly in volts, thus be- 
coming a voltmeter. A few of the more general forms 
will be here shown. 



Fig. 52 shows Kelvin's vertical 




Fig. 53, 



electrostatic voltmeter. As may be clearly seen it con- 
sists of a vertically arranged quadrant electrometer with 
one pair of quadrants deleted and used idiomatically. In- 
stead of a mirror it has a pointer playing over a scale 
graduated directly in volts. These instruments may be 



110 STATIC ELECTRICITY. 

made to operate on from 400 to 20,000 volts, any one in- 
strument having for its maximum reading ten times its 
minimum reading; for instance, from 400 to 4,000 volts, 
or from 2,000 to 20,000 volts. These ratings are for di- 
rect current, and should be just about halved for alter- 
nating potentials, because of the greater sparking dis- 
tance of the latter for a given effective voltage. 

For great accuracy and permanency, such as is required 
of voltmeters for ultimate standards in laboratory work, 
Kelvin's multicellular voltmeter may be used. These are 
made with either vertical or horizontal scales. As shown 
in Fig. 53, they consist of a number of modified quad- 
rant electrometers, arranged coaxally, the vanes all being 
connected together. This gives considerable torque and 
allows substantial and accurate construction. The ranges 
of these instruments are from a minimum to< a little over 
twice that for a maximum, and they can be made to oper- 
ate from 30 to 2,400 volts. 

The instruments thus far described have been labora- 
tory instruments. A type of commercial switchboard elec- 
trostatic voltmeter is shown in detail in Fig. 54, This is 
the Stanley static station voltmeter. The principle is 
again that of the quadrant electrometer. The vanes A A 
are partial cylinders, as are also the fixed quadrants. 
The instrument is idiostatic, the vanes A A and the quad- 
rants B B being all connected to one side of the circuit, 
while C C are connected to the other. The movable mem- 
ber is mounted on knife-edge jewels and is controlled by 
the spiral spring 8. Two fans, F F \ attached to the vane 
and inclosed in the damping boxes D D make the instru- 
ment dead beat. These voltmeters range from 750 to 
1 ; 400 volts, and from 1,400 to 2,800 volts, 



INSTRUMENTS. 



Ill 



The advantages of this type of instruments for switch- 
board use are: permanence of calibration, quick action, 
no loss of current, requires no step-down transformer, free 
from effects of frequency or wave shape, unaffected by 
external fields, and has no temperature error. 




Fig. 54. 



While the use of static instruments is in most respects 
far preferable to that of electromagnetic instruments on 
high tension service, the former are open to one serious 
objection, and that is the tendency of the parts to discharge 
across the air gap between them, giving rise to sparks det- 



112 



STATIC ELECTRICITY. 



rimental to the instruments. By making the gaps wide 
enough to prevent the possibility of this happening, the 
sensibility of the instrument is seriously impaired. To 
overcome this trouble Prof. Elihu Thomson has devised a 
method of inclosing the instrument in a glass tube or 
bulb, exhausted to an extremely high Crookes vacuum. 
By this means the sparking distance for a given voltage is 
greatly reduced, thereby increasing the sensibility and de^ 
creasing the delicacy of the instrument. Fig. 55 shows 
such an arrangement. 




Fig. 55. 



46. Electrostatic Ground Detectors.— The electrostatic 
electrometer principle may be applied to the detection 
of grounds on transmission lines. Fig. 56 shows the 
Stanley static ground detector. This is essentially a differ- 
ential static voltmeter, composed of six fixed metal vanes, 
two primary and four secondary, and a movable vane of 
sheet aluminum carrying a pointer. The movable vane 
and pointer are mounted on a shaft and held centrally be- 
tween the fixed secondary vanes by two sapphire jewel 



INSTRUMENTS. 



113 



bearings, the whole being controlled by a spring. The 
diagonally opposite fixed secondary vanes are connected, 
and each pair is indirectly charged (statically) from one 
of the fixed primary vanes, which in turn receive their 




Fig. 56. 



charges from the line wires. The movable vane is per- 
manently connected to earth, and is inductively acted upon 
by each pair of fixed secondary vanes. The force of each 
8 



114 



STATIC ELECTRICITY. 



pair being equal and opposite the pointer takes a medial 
position. This instrument is thus hetrostatic in its oper- 
ation. If, however, one line wire becomes grounded the 
movable vane receives a charge the same as that of the 
fixed vanes belonging to that wire, and the arrangement 
becomes idiostatic and the movable vane will have a ten- 







I 



Fig. 57. 



dency to lie under the other pair of fixed vanes. This 
may be made to indicate not only a ground, but on which 
wire such ground is. Since there is no electrical connec- 
tion between the line wires and the instrument save to the 
two primary fixed vanes, and as these are imbedded in 
the base of the instrument; it would be impossible, no mat- 



INSTRUMENTS. 



Hi 



ter how badly the instrument was out of order, for a short 
circuit to occur between the lines through the instrument. 
The General Electric Company have devised a ground 
detector for use on three phase circuits. Fig. 57 a'ives a 
view of this instrument, and Fig. 58 shows the electrical 




connections. Here there are three fixed segments corre- 
sponding to the quadrants of previous instruments, and 
three aluminum movable vanes, with pointers attached, ar- 
ranged as shown. Light counterweights are used to hold 
the pointers normally at the zeros. Each of the fixed seg- 
ments is connected to a line wire and all the movable vanes 
are connected to the ground, as shown in Fig. 59. ^Vhen 



116 STATIC ELECTRICITY. 

no ground exists the needles point toward the center. 
When a ground occurs on one of the lines the two adjacent 
needles are deflected toward the segment connected to> that 
line. Should a ground occur on two lines simultaneously, 
the needle between the segments belonging to the grounded 
lines will point to the line having the least resistance 
ground, and the remaining two will be deflected toward 
the grounded segments. If a vane became displaced and 
touched a segment it would in itself constitute a ground, 
and if it touched two segments at once, it would short- 
circuit the lines and ruin the instrument. To guard 
against this contingency the segments are connected to 
the lines only through high carbon resistances, as are also 
the vanes to the earth. These are shown in Fig. 58. 



CHAPTER VI. 

HIGH POTENTIAL STATIC GENERATORS. 

47. Friction Machines. — It was but natural for the 
early experimenters to seek for some more satisfactory 
means of securing electricity than the laborious process of 
rubbing a glass or vulcanite rod with cloth or fur, which 
process was slow, and by its very nature intermittent. As 
might be expected the first step was to devise a means 
whereby the rubbing might be mechanically and contin- 
uously performed. 

The most primitive machine for this purpose was prob- 
ably that attributed to von Guericke, which consisted 
merely of a sphere of sulphur mounted on a spindle, with 
means provided for revolving it. If the hand were held 
against this sphere while in rotation, a slight charge 
would be imparted to the body. Succeeding steps replaced 
the sulphur sphere with one of glass, substituted a leathern 
cushion for the hand as a rubber, simplified the construc- 
tion by changing the spherical revolving member to a 
cylindrical 'one, and added a loose metal chain which, 
hanging against one side of the revolving cylinder, could 
conduct the charge away. 

A practical form of the friction machine used to a lim- 
ited extent at present in laboratory demonstration con- 
sists of a horizontal glass cylinder, mounted on metal shaft, 
equipped with crank and handle, and suitably mounted in 
bearings on strong supports. On one side of the cylinder, 
and pressing against its convex surface, is a pad of leather, 

117 



118 STATIC ELECTRICITY. 

covered with a suitable amalgam, and having an apron or 
flap of silk attached, which lies over the upper half of 
the cylinder. The rubbing pad is usually mounted on an 
insulating support, but if it is never desired to collect the 
negative charge which accumulates thereon, this refine- 
ment may be omitted. The direction of rotation of the 
glass cylinder is such as to draw the silk apron the more 
rightly to the surface, thus considerably increasing the 
area exposed to contact, and increasing the amount of 
charge. On the side of the cylinder opposite the rubber 
is arranged a horizontal metal comb with the teeth point- 
ing toward the axis of the cylinder. The back of this comb 
feeds into a large metal sphere or round ended cylinder, 
which is customarily called the " prime conductor." 

The action of the comb is this: The friction, or more 
properly the contact between the rubber and the glass, 
causes electrical action such that a negative charge accu- 
mulates on the amalgam-coated surface of the rubber and 
a positive charge on the glass. The surface of the glass in 
revolving brings this charge underneath the comb, where 
it acts by influence on the prime conductor, attracting a 
negative charge to the teeth of the comb, and repelling a 
positive charge to the further extremity of the prime con- 
ductor. Now, as stated in section 14, the pointed teeth of 
the comb are but poorly adapted to retaining a charge, and 
the negative charge escapes from them in the form of 
an electric wind which blows directly on the cylinder. 
The negative electricity thus carried to the surface of the 
cylinder unites with the positive charge already there, 
leaving the surface approximately neutral as it approaches 
the rubber for a repetition of the cycle. Thus increments 
of free positive charge are being added to the prime con- 



STATIC GENERATORS. 119 

ductor as elements of the surface pass the comb. The 
quantity of electricity that may thus he accumulated in 
the prime conductor depends upon the insulation of the 
latter. For the electrostatic capacity of the prime con- 
ductor and its insulation from surrounding objects are 
sensibly constant, hence, as the amount of charge increases, 
so also must the potential increase. But the leakage cur- 
rent, which follows Ohm's law, will increase as the po- 
tential increases ; so it is evident that a point will be 
reached where the leakage current carries away from the 
prime conductor as much electricity as the revolving cyl- 
inder brings to it. This leakage of electricity occurs not 
only over the surface and through the substance of the 
insulating supports of the prime conductor, but a passage 
of electricity from the conductor directly into the air itself 
occurs, which gives rise to the " brush discharge," which is 
treated of in a subsequent paragraph. In using this in- 
strument the rubber, which because of its amalgam coating 
is a good conductor, is generally connected to ground by a 
chain or other convenient means, to carry off the negative 
charge which tends to accumulate thereon, thus leaving it 
in a condition to more strongly excite the electric action. 
This amalgam with which the rubber is coated serves a 
double purpose. As just mentioned, it renders the rubbing 
pad much more conductive than it would be if simply 
covered with leather, and it offers as a friction surface a 
substance between which and glass more vigorous electric 
action takes place than between leather and glass. In the 
" Electrostatic Series," given in section 6, it is seen that 
the interval between glass and metals is greater than that be- 
tween glass and the hand, the latter being, electrostatically, 
about the same as leather. A simple amalgam of tin and 



120 STATIC ELECTRICITY. 

mercury will answer the purpose, A more satisfactory 
compound, known as Kienmayer's amalgam, consists of 
equal parts by weight of tin and zinc melted together, to 
which, while still molten, is added twice their weight of 
mercury. These amalgams are usually applied to the 
rubbing cushions through the agency of some thick grease. 

A more usual form of the friction machine has for the 
revolving element a circular disc of glass instead of a 
cylinder. This form of construction, besides being much 
cheaper, since a perfect glass cylinder is difficult to make, 
allows the use of two or more sets of rubbers and allows 
both sides of the glass to be used, thus greatly increasing 
the surface exposed to friction without increasing the size 
or weight of the machine. Furthermore, the distance over 
the surface between charges of opposite sign is greater, 
tending to reduce losses by internal discharge, or leakage 
over the surface of the revolving member. Friction ma- 
chines have been entirely superseded for all but labora- 
tory and demonstration purposes by the influence ma- 
chine, the simplest form of which is the electrophorous, 
described in section 10. 

48. The Toepler Machine. — Toepler began the experi- 
mental construction of influence machines about 1865, 
before which date little had been done along this line, 
though the first so-called " doubler " was invented in 1786. 
A doubler is simply an elaboration of the electrophorus 
(§ 1.0) such that the charge influenced on the cover is not 
discharged to earth but to a prime conductor. The cover 
is then sent back to the electrified hard rubber, charged 
by influence again, and this second charge discharged into 
the prime conductor. As this operation is repeated it is 
evident that the quantity of electricity on the prime con- 



STATIC GENERATORS. 



121 



ductor will be increased, and at each increase its poten- 
tial will also be raised. 

The Toepler machine, as now constructed, is shown in 
sketch in Fig. 59. A fixed vertical glass plate, usually 
circular for convenience, has a hole in its center through 
which a horizontal revolving axis can work between bear- 
ings, one on each side of the fixed plate. Mounted on the 




Fig. 59. 



axis in front of the fixed plate is a revolving glass plate 
of a diameter somewhat less than that of the fixed* plate. 
The distance of the revolving plate from the fixed plate is 
made as small as is practicable. On the front of the ro- 
tating plate, near the periphery and at equal intervals, 
are pasted small circular discs of tinfoil, called carriers, 
each disc having at its center a small protruding metallic 



122 STATIC ELECTRICITY. 

button. This button allows of metallic connection be- 
tween the moving carriers and the fixed brushes without 
unduly wearing the former. On the back of the station- 
ary plate and at opposite ends of its horizontal diameter 
are fixed two pieces of paper, called field-plates or induc- 
tors. These field-plates are of somewhat segmental shape, 
as shown by the dotted lines in the figure, and are made 
of paper, because it is found that this substance, being a 
fail conductor of static charges, is more satisfactory than 
a good conductor, such as tinfoil. As shown in the figure, 
from the end of each field-plate which the moving car- 
riers first approach, a conducting arm is bent so as to hold 
a soft wire brush over the front of the revolving plate in 
such position that the brush just makes contact with the 
metal buttons on the carriers as the latter move under the 
former. These are called the appropriating brushes. Two 
more brushes held in place and in electric connection by a 
diagonal metallic bar, called the neutralizing rod, touch the 
buttons just as the carriers are passing beyond the end of 
the field-plates. In front of the rotating plate and oppo- 
site the centers of the field-plates are two metallic combs, 
teeth pointing to the plate, and connected to sliding dis- 
charging rods, called electrodes, the ends of which can be 
separated by any required distance. 

The action of the machine is as follows: Neglect for 
the moment the discharging combs and rods. Suppose a 
small negative charge to rest on the left hand field-plate. 
This will attract a small positive charge on the under side 
of a carrier that happens to be over it. A similar negative 
charge will be repelled to the front side of this carrier. 
Now assume the disc is revolved in the proper direction. 
As the carrier passes under the neutralizing brush it is 



STATIC GENERATORS. 123 

still under ;the influence of the field-plate, therefore the 
negative charge on the front of the carriers is repelled into 
the neutralizing rod and disposed of, as will be shown 
later. As the carrier passes beyond the influence of the 
field-plate the positive charge bound on the under side of 
it will become free and, as the corresponding negative 
charge has been disposed of by the neutralizing rod, the 
carrier will have a free positive charge overlying its 
whole surface. When the carrier passes under the next 
brush a part of this charge will be communicated to the 
right hand field-plate, thus giving it a charge of opposite 
sign to that of the left hand field-plate. The carrier now 
undergoes the second half of the cycle, being charged and 
discharged as in the first half, save that all the signs are 
changed, a positive charge being sent out into the neutral- 
izing rod. This charge unites with an equal negative 
charge coming in from the left hand end, as mentioned 
before, and thus the rod is kept neutral without the ne- 
cessity for a ground connection. The carrier is now on its 
way toward the left field-plate with a free negative charge, 
a part of which it gives up to the plate as it passes under 
the next brush, thus increasing the original charge. This 
cycle is repeated for each carrier at each revolution, and 
it is thus clear that the field-plates will soon be charged to 
so high a potential that the amount of electricity lost from 
the field-plates by leakage equals that brought to them by 
the carriers. The function of the metallic carriers and 
the brushes is simply to keep the field-plates as highly 
charged as possible. 

Now consider the discharging combs. Imagine the ends 
of the discharging rods to be in contact. While the ma- 
chine is building up these combs and rods have no func- 



124 STATIC ELECTRICITY, 

tion. When, however, the field-plates have their poten- 
tial raised to a certain point, their influence on the metal 
of the discharging apparatus becomes so strong — the 
two intervening plates of glass offering no obstacle — 
that the electricity attracted to the points of the combs 
can no longer rest on them but is torn off in the form 
either of an electric wind (§ 14) or a brush discharge. 
The electricity thus transferred from the combs rests on 
the front surface of the revolving plate, and is not dis- 
sipated to any extent, as the plate is a non-conductor. Kow 
positive electricity is fixed on the rotating plate above 
the left hand or negative field-plate. As the machine re- 
volves this electricity is brought around under the right 
hand comb, from which negative electricity is streaming. 
This latter unites with the positive already on the rotating 
plate, leaving the plate approximately neutral and in con- 
dition to repeat the process. This constant discharge from 
the combs implies a transference of electricity along the 
electrodes, and this is the fact. The circuit is curious in 
that it is not a complete conducting circuit, but through 
one part of it the electricity is carried by mechanical 
means. Beginning at the face of the revolving plate, just 
under the first comb, the circuit can be traced as an ordi- 
nary conducting circuit up through the brush discharge, 
through the comb and electrode, through the other elec- 
trode, comb, and brush discharge to the face of the plate 
under the second comb. Here the circuit ceases to be con- 
ducting, the electricity being carried back to the starting 
point by the rotation of the plate. It is in this part of 
the circuit that the energy is supplied that keeps the elec- 
tricity moving and does the work that is expended in 
sparks or vacuum tubes or other apparatus that may be put 



STATIC GENERATORS. 125 

in circuit. The action of this circuit under various con- 
ditions of resistance and capacity are treated of in a sub- 
sequent paragraph. 

This machine, or slight modifications of it, are some- 
times called Toepler-Holtz or Voss machines, because of 
modifications made by others than Toepler himself. As the 
inventor left the machine it differed from the one just 
described, in that the fixed plate consisted of two separate 
pieces of glass, one to the right and one to the left. The 
space between these was sufficient to accommodate the re- 
volving axis. In the original machine also the neutralizing 
bar and brushes were lacking, and the appropriating 
brushes were fixed at the trailing ends rather than the 
leading ends of the field-plates. The field-plates them- 
selves were of tinfoil. With this arrangement the charges 
repelled to the front surfaces of the carriers were added 
to those on the repelling field-plates. This left the two 
halves of the machine independent save for the discharge 
circuit, a condition which lends itself readily to unbalanc- 
ing the system. It is found that the maximum effect can 
be obtained from a machine when the field-plates are 
charged to equal potentials of opposite sign. Toepler found 
that his machine operated much better if he removed one 
tooth from each of the combs and substituted a brush that 
would touch the carriers. The effect of this was to equal- 
ize the potentials. It was almost essential that this ma- 
chine be started with the discharge rods in contact, to aid 
in the building up, while in the improved form it is some- 
what better to have the rods widely separated. 

The Toepler machine is very satisfactory, in that it can 
be started in any weather and without excitation, it being 
always the case that one field-plate is at a slightly higher 



126 STATIC ELECTRICITY. 

potential than the other. Even if it were not the mere 
agitation of the air in the neighborhood would cause 
enough difference to enable the machine to start. 

49. The Holtz Machine. — This machine, which is per- 
haps the best known of the influence machines, consists of 
a fixed plate and a revolving plate as shown in Fig. 60. 
The former is slightly the larger in diameter and is pierced 
by three holes, a circular hole in the center to accommodate 
the revolving axis, and two " windows " near the opposite 
ends of a diameter and of a shape somewhat as shown. 
These windows are sometimes replaced by deep notches 





Fig. 60. 

running in from the circumference, a form that seems 
equally good and is much easier to cut. On the back of 
the fixed plate and at the side of each window are pasted 
the field-plates. These field plates are usually of paper 
and are of such circumferential length as to subtend from 
10° to 60° of central angle, between these limits their 
length seeming to make little difference in the operation 
of the machine!. One, or sometimes more, rather blunt- 
pointed projections from each of the field-plates are 
brought through the windows and bent so as to point 
toward but not touch the revolving plate. The revolving 
plate is mounted concentric with the fixed plate and as 



ST A Tl C (J EX ERA TORS. 



127 



near to it as is practicable, the less the clearance being the 
better the operation of the machine. There are no carriers 
on the revolving member, this being simply a shellaced or 
lacquered glass disc. As shown in Fig. 01 two collecting 
combs are mounted in front of the machine, one opposite 
each field-plate, with teeth pointing toward the revolving 
plate and set a little way back from the edge of the window. 




Fig. 61. 



It is stated that with these combs set considerably further 
back from the edges of the windows tinfoil field-plates can 
be satisfactorily used. These combs are connected to the 
usual discharging rods or electrodes. This equipment 
completed the theoretical construction of Holtz's original 
machine, but he subsequently added, and all modern ma- 
chines are equipped "with, a neutralizing rod fitted with a 
comb at each end, the rod being held diagonally in such a 
position that the combs are about over the hinder or trail- 



128 STATIC ELECTRICITY. 

ing edges of the field-plates. The direction of rotation of 
the moving plate musl be toward the projecting points of 

the field-plates. 

The operation of the machine is as follows: The elec- 
trodes are pnt in contact and the disc is rapidly rotated. 
A charge of Borne magnitude is then imparted to one of 
the field-plates. This i- usually accomplished by briskly 
rubbing n sheet <>f paper with :i cloth or fur and then plac- 
ing it on the back of the fixed plate over one <>( the field- 
plates, where it will 9tay of itself if it is sufficiently well 
charged to start the machine. Say tin- aegative charge i^ 
imparted to the Left hand field-plate. Tin- charge repels a 
negative charge in the collecting apparatus and attracts a 
positive charge to the teeth of the comb. The charge on 
the field-plate musl be sufficiently strong to tear this p «i- 
tive electricity off from the comb in the form of an el< 
wind, or the machine will not build up. If the wind does 
blow upon the revolving plate the front surface of the latter 
becomes positively charged, and a- the plate revolves this 
charge is brought around over the projecting point of the 
right hand field-plate. TTere it attracts negative electricity 
from the right hand field-plate which, blowing off the point, 
becomes bound on the under side of the revolving plate, 
the glass of the latter acting as the dielectric of a condenser 
separating two mutually attracting charge?. The loss of 
this negative electricity leaves the right hand field-plate 
positively charged. This plate now draws a negative wind 
from the right comb which, uniting with the positive 
charge already on the front of the revolving plate, 
neutralizes the latter, thus freeing the negative charge on 
the under side of the revolving plate, and then leaves an 
excess of negative electricity on the upper side as well. 



129 

A 

■ 
■ 
n until ll 

I • 

I • | 

if the 

Fep reversals of 

jump tl- 

I the 

■ 

the 

- 
9 



130 STATIC ELECTRICITY. 

which, as soon as the potential of the machine begins to 
fall, start discharging in a direction opposite to that in 
which they were charged. The presence of the neutraliz- 
ing rod, offering as it does a low resistance path irrespective 
of the position of the electrodes, assures the satisfactory 
neutralization of the revolving plate under any conditions. 

The Eoltz machine is Bomewhal more powerful than the 
sehPstarting machines of similar size bul is not alwa; 
satisfactory because of it- dependence on favorable atmos- 
pheric conditions. Sometimes it is impossible to start it 
in the open room at all, and often it is only after a long 
period of warming, drying, and rubbing that it will work. 
In the latter cases it seems as if the internal discharges of 
the machine improve its condition, for after one- being 
started it, usually runs without further trouble, and may 
even he stopped and easily Btarted again within a re 
able time. The only way, however, to insure any certainty 
oi operation is to inclose the whole machine in a glass case 
as nearly air-tight and dust-proof as is practicable. In this 
case should be put a -hallow dish full of concentrated sul- 
furic acid to absorb the moisture of the air, and another 
fnll of linseed oil which seems to absorb the ozone given 
of! during operation. The presence of ozone lowers the in- 
sulation of the air and decreases the efficiency of the ma- 
chine. Some Holtz machines are even incased in such a 
manner that the whole machine and the air surrounding it 
may be warmed by a flame in an iron oven in the lower 
part oi the case. 

Several modifications in the construction of these ma- 
chines are sometimes observed. The possible use of tinfoil 
field-plates and a notch instead of a window for the point 
of the field-plate to project through have already been 



I 

I* the 

: 1 

- 

II '• • ] 
will 

and • 



132 



STATIC ELECTRICITY. 



respectively and their outer coatings connected together. 
The jars will at first discharge but the machine is thereby 
charged to such an extent as to become operative and the 
jars are then charged to the full capacity of the machine 
in the same sense that they were at first. If the machine 




Fig. 62. 

were not equipped with a neutralizing rod the electrodes 
would have to be put in contact immediately after the jars 
were connected else the whole charge would be lost. 

Holtz made many other forms of machine, some with 
two opposite revolving plates, some that were not regenera- 
tive, that is, did not keep their own field-plates charged, 
but these are now only of historical interest. 

50. The Wimshurst Machine ■ Unlike the foregoing 

machines, the Wimshurst has no fixed field-plates. As 



STATIC GENERATORS. 133 

shown in the sketch (Fig. G2), two similar, oppositely 
rotating glass discs are concentrically mounted as near to- 
gether as is feasible. On the outer face of each of these 
plates are attached a number of sector-shaped carriers of 
metal or foil, radially arranged and equally spaced. A 
neutralizing bar with brushes is provided for each plate, 
such bar being set at an angle of about 45° from the col- 
lecting combs measured in the direction of the rotation of 
the plate to which it appertains. The collecting combs are 
bent, horseshoe-like, to allow the teeth to project toward 
the outer face of each plate and to cover the full radial 
depth of the metal sectors. The electrodes are customarily 
mounted on pivots so the distance between them may be 
conveniently varied. It is usual to have the knobs on the 
end of the electrodes of unequal size, as it is found that a 
greater length of spark can be obtained with this arrange- 
ment. 

The action of this machine should be readily under- 
stood because of its similarity to that of the Toepler ma- 
chine. Consider for a moment the carrier (Fig. 62) that 
is being touched by the upper front neutralizing brush. In 
back of this and on the other plate is another carrier which 
we will assume to carry a small negative charge. This last 
will act like the first field-plate of the Toepler machine 
and leave the front carrier with a positive charge after it 
has passed from under the brush. In the meantime a 
negative charge will have been repelled down the neutraliz- 
ing bar, leaving the diametrically opposite carrier on the 
front plate negatively electrified. As now our original 
carrier is rotated till it comes opposite the upper back 
neutralizing brush it will act as a field-plate to the carrier 
under it on the back plate, thus giving the latter a negative 



134 STATIC ELECTRICITY. 

charge. Meanwhile the negatively charged lower front 
sector will have influenced a negative charge on a back 
sector. This process is repeated for each carrier, the ten- 
sion of the charges increasing at each repetition until the 
limit set by the leakage of the machine is reached. It will 
be observed that each plate is carrying positive charges 
toward the right collecting comb and negative charges 
toward the left. These combs as it is frequently put " col- 
lect " these charges, the more proper statement being that 
the charges on the carriers attract and are in part neutral- 
ized by brush discharges drawn from the combs. 

This machine is self-starting in practically any kind of 
weather if the neutralizing brushes are clean and make 
good contact simultaneously with diametrically opposite 
carriers. As the number of sectors is decreased and the 
space between them correspondingly increased the length 
of the sparks obtained is increased but their number is de- 
creased. This is a result of the lessened internal leakage 
due to the wider spacing. If the number of carriers is 
reduced beyond a certain limit, about six per plate, the 
machine ceases to> be self-exciting but will work well after 
being properly charged. It is usual to start the machine 
with the electrodes in contact, but this precaution can be 
omitted as the discharge circuit plays no necessary part 
in the upbuilding of the charges on the sectors. Unlike 
the Holtz and the Toepler, the Wimshurst machine never 
reverses its polarity, even when the electrodes are separated 
much beyond the sparking distance. Neither does it loose 
its charge if left running with the electrodes in contact. 

51. Static Machines as Motors. — Any of the machines 
described in the foregoing sections can be operated as a 
motor if supplied with electricity of sufficiently high ten- 



STATIC GENERATORS. 135 

sion. If one Holtz machine be regularly driven as a 
generator and its electrodes be connected to the electrodes 
of a second Holtz machine and the electrodes of both ma- 
chines be well separated so that sparks do not jump the 
gaps, the second machine will start up and run in the back- 
ward direction, that is, away from the tongues of the field- 
plates. 

Such a combination as this has been operated by an 
electric motor whose mechanical output at any given elec- 
trical input had been previously determined. The me- 
chanical output of the second Holtz machine, running as 
a motor, was measured by a delicate Prony brake, while 
the input to the electric motor was measured by a watt- 
meter. After allowing for the losses in the electric motor, 
but not allowing for losses in the driving belt, the speed- 
measuring device nor for any electrical losses between the 
two static machines, the combination of Holtz generator 
and Holtz motor showed an efficiency of some 7 per cent. 
Assuming the machines to have the same efficiency this 
would indicate an efficiency in each machine of V.07 or 
27 per cent. 

52. The Dropping Generator. — The three machines de- 
scribed in the previous part of this chapter illustrate the 
three principal types which are the foundation for most of 
the modern influence machines. In practice many modifi- 
cations are introduced, one of the principal ones being the 
running of a number of plates in multiple to secure a 
greater volume of discharge. Multipling the plates also 
slightly increases the spark length for the same diameter 
plates. Some of the salient features described above are, 
however, generally to be found in all static generators. 

A high tension electric influence machine of an entirely 



136 



STATIC ELECTRICITY, 



different character is the dropping generator due to Kelvin. 
This machine which is of more theoretical interest than 
practical utility is easily described. From a 



common 



reservoir two streams of water are led. (Fig. 63..) Each 
stream is allowed to fall vertically downward through a 
hollow insulated metal cylinder 3 or 4 inches long and 
about li inch in diameter. These streams are so adjusted 




Fig. 63. 



that they break into drops when about half-way through 
the cylinders. The drops are collected beneath the cylin- 
ders by insulated metal funnels and the water is allowed 
to fall in a spray from the funnels to the waste receptacle. 
As these funnels are to become the oppositely charged parts 
of the apparatus, a solid stream from them to the waste is 
impracticable because of the short-circuiting effect. 



STATIC GENERATORS. 137 

Let us assume that one cylinder is slightly charged in 
the negative sense. As the drops that leave it break off 
from the solid stream they are positively electrified, the 
corresponding negative charge being repelled up the stream 
to the reservoir. These drops are collected in the funnel, 
and as there can be no charge within a conductor, the drops 
immediately give up their charges which are transferred to 
the exterior of the funnel. The drops then issue from the 
bottom of the funnel approximately neutral. This funnel 
is kept in metallic connection with the second cylinder, 
thus the latter becomes positively charged. Then the sec- 
ond funnel has added to it constant increments of negative 
charge. This funnel is likewise connected by wire to the 
first cylinder, thus augmenting the negative charge thereon. 
The limit to the increase of potential obtainable by this 
apparatus is set by the leakage of charge over the surface, 
as in foregoing cases, but here it is easier to improve the 
insulation, as the various parts may be mounted on long 
glass rods or supports, and there are no moving parts save 
the water. It is stated that 7,000 volts — electromagnetic 
system — can readily be obtained by this machine. 

As the drops issuing from either of the cylinders are 
highly charged they are strongly self-repellant, and as the 
funnel beneath waiting to catch them is similarly charged, 
tending further to repel the drops, it usually happens that 
only a portion of the water finds its way into the desired 
receptacle, the rest being scattered throughout an uncom- 
fortably wide range. 

53. Discharges of the Static Machine. — The charged 
terminals of an electric machine or a battery of Leyden 
jars charged by it may be .discharged in at least three 
rather distinct w^ays. The first is by simple metallic con- 



133 STATIC ELECTRICITY. 

duction between the charged bodies, as when the electrodes 
are united by a wire. The only phenomenon connected 
with such a discharge is the generation of heat in the con- 
ductor which is strictly according to the PR law of 
dynamic electricity. The current, however, is not uniform 
and a more adequate expression for the heat generated — 
assuming all the energy of the discharge is transformed 
into heat — is 

V T ■ 

B. = calories, 

where J is the mechanical equivalent heat, expressed in 
ergs per calorie. (J = 42 x 10 6 .) For it was shown in sec- 
tion 34 that 2 Q V represents the total energy of a charge. 
The energy thus set free is usually too insignificant to be 
detected, but discharges from large machines equipped 
with heavy condensers have been made to heat a consid- 
erable length of wire, and to fuse pieces of tinfoil. 

In elaboration of an experiment by D'Arsonval, Dr. 
Sheldon has succeeded in lighting to full brilliancy an or- 
dinary 16 candle-power 116 volt incandescent lamp. He 
connected the electrodes of a large IToltz machine re- 
spectively with the inner coatings of two Ley den jars. 
The electrodes were set with a reasonably large spark gap. 
The outer coatings of the jars were connected together 
through an impedance coil consisting of half a dozen turns 
of heavy copper wire around a four-inch glass cylinder. 
From the extremities of the impedance coil wires were led 
off to the lamp. After suitable adjustment of spark gap 
and speed the lamp glowed at full brilliancy. A hot-wire 
ammeter inserted in the lamp circuit showed the lamp to 
be taking its normal current. 

The second method of discharge is the so-called " dis- 



STATIC GENERATORS. 139 

ruptive discharge/' meaning a discharge through the 
agency of a spark. If an influence machine be operated 
without Leyden or other condensers it will be found im- 
possible to cause the discharge to cross anything but a 
small gap. When the electrodes are brought sufficiently 
near together, say within an inch, a stream of fine sparks 
keeps up with such rapidity of succession as to look like 
one continuous thread of light between the terminals. 
This thread does not, however, stay stationary, except in 
the case of particularly short gaps, but cuts out an ever- 
changing zigzag course from one electrode to the other. 
Such a discharge is accompanied by a gentle hissing noise. 
If now a couple of Leydens be added, their inside coatings 
connected to the electrodes respectively and their outer coat- 
ings connected together, the discharge loses its continuous 
character and becomes intermittent. At regular intervals of 
a few seconds what appears to be a single bright spark passes 
accompanied by a sound like the snap of a whip. If now 
the electrodes be separated the intervals between sparks 
become longer and the discharge becomes brighter and more 
noisy. The more jars that are added, and, up to the spark- 
ing limit of the machine, the wider the separation of the 
electrodes, the more crashing the reports become. As the 
spark becomes longer it appears to take on a forked shape^ 
and a photograph of such a spark usually shows many 
forkings. The branches of such a fork invariably point 
toward the negative electrode, or better, more nearly to the 
negative than the positive. Faraday showed by experiment 
that when the electrodes were tipped with spheres of un- 
equal diameters, the longest sparks were obtainable when 
the smaller sphere was made the positive electrode. 



140 STATIC ELECTRICITY. 

The disruptive strength of such a spark is considerable. 
A thick book can be held between the electrodes and a hole 
will be pierced through it. So also can a hole be pierced 
through glass or mica, though the thickness of these 
must not be great and the electrodes must be brought 
close to the surface to be punctured. Frequently the 
puncture cannot be made unless the electrodes are 
made of sharp points instead of the usual spheres. These 
points then tend to concentrate the stress that might other- 
wise be spread over a considerable surface, the positive 
electricity spreading out on one side, the negative on the 
other, the glass or mica functioning merely as the dielectric 
of a condenser. The insulator must also be of sufficient 
size, for the spark will travel a long way over the surface 
to get around the edge rather than pierce the substance. 
The distance the spark will travel in this manner is greatly 
in excess of the distance it would jump in open air because 
of the conductive properties of the surface, which must of 
necessity be somewhat moist and dirty. 

If such a discharge be passed through a piece of paper 
or cardboard, a curious effect can be observed. It will be 
found that the hole is not only burred on one side as it 
would be if it had been pierced with a needle, but is burred 
on both sides. This has been taken as an indication of the 
oscillatory character of the discharge, but the same effect 
would be observed if the stresses in the air are such as to 
pull at the card from each side. 

Prof. S. P. Thompson gives the following formula for 
approximately determining the voltage necessary to start 
p spark across an air gap of I centimeters, the gap in any 
event being greater than two millimeters. 
7-1,500 + 30,000?. 



STATIC GENERATORS. 141 

If the length of the gap is expressed as L inches, the 
same formula becomes 

V = 1,500 + 76,000 L. 

The third method of discharge is by convection. This 
form of discharge can take place in any non-conducting 
fluid, either liquid or gaseous, but the only form of im- 
portance is the discharge in air. If an influence machine 
be well excited and the terminals separated to greater than 
the sparking distance, a discharge will take place from 
each terminal which can be seen in a dark room as a light 
blue luminescence branching out in such shape as to give it 
the name of brush discharge. A hissing noise is always 
heard in connection with this phenomenon. The electric 
wind as described in section 14 is the simplest form of 
convection discharge, but this can only emanate from a 
sharp or pointed conductor. If, as in the case above de- 
scribed, the conductor is finished entirely in curves and 
spheres, such a starting point is lacking and the charge 
seems to take the form of a continuous spark for a short 
distance from the conductor. This spark branches and 
radiates into numberless paths, each of which becomes a 
moderately good conductor and conveys the charge to a 
point where it can be transferred to the air particles as in 
the electric wind. A difference can be observed in the 
brush discharges from the positive and from the negative 
poles, the brush at the positive pole being larger and more 
bushy than the other. 

Spectrum analysis of the brush discharge shows the 
presence of volatilized metal of which the conductor is com- 
posed. In the disruptive discharge if the terminals be of 
different metals, traces of both metals can be found in the 
spark. This indicates the actual consumption of the metal 



142 STATIC ELECTRICITY. 

of the electrodes and accounts for the pitting observed on 
the surface of the discharging knobs after they have been 
in service some time. A pair of knobs which when smooth 
and well polished will give a disruptive discharge across a 
certain gap will often give a brush discharge when spaced 
the same but with a surface roughened and pitted. There- 
fore, to get the maximum length of spark from a machine 
the electrodes should be kept in a bright condition. 

It is found that the resistance of air to the passage of 
electricity decreases as the pressure decreases until a cer- 
tain point is reached, after which it increases with the de- 
crease of pressure till at the nearest approach to a vacuum 
that can be practically attained the resistance is much 
higher than at atmospheric pressure. The discharges 
which occur through tubes exhausted to these vacuums are 
believed to be due to convection. They give rise to many 
interesting and unexplained phenomena, Geissler effects, 
cathode rays, Roentgen rays, Becquerel rays and others, a 
discussion of which is entirely without the scope of this 
present work. 



APPENDIX. 



A. Dimensions of Units. — Nearly, if not quite all, the 
quantities dealt with in exact science can be expressed in 
terms of three fundamental units. The choice of these 
three is to a great extent a matter of convenience, for there 
are many combinations of units that might be employed. 
For instance, all units could be referred to a fundamental 
unit of mass, one of energy, and one of density ; or to one 
of mass, one of velocity, and one of time. The units act- 
ually chosen for use in the c. g. s. system, which is the only 
system that has any pretension to being uniform and in- 
ternational, are as follows : 

A definite length, called a centimeter, 

A definite mass, called a gram, and 

A definite interval of time, called a second. 

These units were chosen chiefly because they admitted 
of accurate and relatively easy comparison with other quan- 
tities of the same kind; because permanent standards of 
them could be made (except of time. This depends upon 
the uniformity of the earth's rotation on its axis) ; and be- 
cause the standards do not vary with gravitation nor 
barometric pressure. 

These three units are called the fundamental units. All 
others are called derived units. All units based directly 
upon the fundamental units, without the intervention of 
reduction factors, are called absolute units. The simplest 
of the derived units is that of area. We could of course 
assume some unit tor this, the acre for instance, but this 
would interfere with the uniformity of the system. We 

143 



144 STATIC ELECTRICITY. 

therefore consider an area as a length times a length. 
Letting A, U , and L" stand respectively for the area, one 
length and the other length, 

A = L' L". 
To get the unit area according to this system we must let 
L' and L" each equal unity, dispensing with the need of 
the indices. Then 

A = LL = L 2 
and we say that the dimension of the unit of area is L 2 . 
The dimension of the unit of volume is an area times a 
length, 

7 = L 2 L = L 3 
and we say the dimension of the unit of volume is L 3 . 

A velocity, whether expressed as miles per hour, feet per 
minute, centimeters per second, or otherwise, cannot be con- 
sidered as anything but a certain distance covered in a 
certain time unless we fall back on other than our funda- 
mental units. (An acceleration times a time would give a 
velocity.) We therefore say that the dimension of the unit 
of velocity is a distance divided by a time, or 
v = LT 1 * 

Acceleration is denned as the increase in velocity per 
unit of time. Hence the dimension of the unit of accelera- 
tion is a velocity divided by a time, thus 
a^vT 1 -LT" 2 

The unit of force is denned as that force which acting 
on the unit of mass for a unit of time generates in it a unit 
of velocity. Evidently the force varies directly as the 
mass, directly as the velocity, and inversely as the time. 
The dimension of this unit will then have a mass and a 
velocity in the numerator and a time in the denominator, or 
/ = v MT' 1 = MLT- 2 



APPENDIX. 145 

According to another definition the unit of force is that 
force which will impart a unit acceleration to a unit mass. 
According to this definition the dimension of the unit is 

/ = aM = MET 2 
which is the same as before. 

So with any unit, not only these simpler mechanical 
ones, but also with the more complex magnetic and elec- 
trical ones ; by referring to its definition, the dimension of 
any unit can be written in terms of L, M, and T, or in 
terms of other units which in their turn have dimensions 
consisting solely of L, M, and T. 

In Table A, which is in part adapted from S. P. 
Thompson's Electricity and Magnetism, is given in con- 
venient form for reference, data concerning the various 
units used in the study of electricity. The dimensions of 
the electrostatic, the magnetic, and the electromagnetic, 
called also the electrodynamic, systems of units are given, 
together with short definitions of each sufficient to show 
how its dimension is derived from one or more units whose 
dimensions have previously been given. Where the abso- 
lute units have corresponding practical units, the names of 
the latter together with their ratio to the absolute or c. g. s. 
values are given. 

A few examples of the less obvious derivations of dimen- 
sions will now be given. For instance, to determine the 
dimensions of the electrostatic unit of quantity of elec- 
tricity. It was shown in section 12 that the force between 
any two quantities varied directly as their product and in- 
versely as the square of the distance between them. That 
is, if proper units be taken, 

f -t£ 



146 



STATIC ELECTRICITY. 



J S H 



th|(S* th|9» 



H H H 



be « 
X + 



bC 

c 



*? X 



bC 

a 



,T3 an to 



to 



S H 



c3 o 

< > 



£ *> 



R| 



a 
•I- 

o 



a 



a 



IV •+-> +J 



5 S « £ 

JT, O c3 c 

O* P-< U W 



ti 



II 



^ s> e <s 



o» U O fe, 



APPENDIX. 



147 




cole* H®* 

2 j 

3 S 



H** H*» eolN , 
iho* ^ He* 



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X 



c x o 

(u s 3 

'O c U 

be be , 

ctf sJ •• 




be j-* 






^ £q &-<* e- 



u -< o 



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73 ^ 

&H £ W 



c_i H°> Ch e* 

J 'HO* K-J H H 

S S J 



J >J 


a 


3 


T3 

a 




c! 




Sr* 




^ 




1 


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eQ 



s 
s 



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o 

is a 

33 2 
■8-32 



3 .3 

S3-? 9 

^3 



O* O PS O Ph^§ 



U«o»s k o a, ^ fe 



148 STATIC ELECTRICITY. 

To get unit conditions the two quantities must be equal and 
their product can be written Q 2 , then 

Q* = /P. 

Now the dimension of a force is MLT" 2 and of a length 
is L. So we may write for the dimension of the electro- 
static unit of quantity 

Q = Vf¥= M*" V T" 1 x L = M* L* T~ l . 

An analogous line of reasoning may be followed in get- 
ting the dimension of the unit magnet pole. 

The electrostatic potential at any point varies directly 
as the work required to bring a quantity of electricity to 
that point from an infinite distance and inversely as that 
quantity. So we can write 

V= — = M^I>T\ 

Q 

The difference in electromagnetic potential between the 
ends of a conductor moving through a magnetic field varies 
directly as the number of lines of force cut and inversely 
as the time taken to cut them. The number of lines of 

1 3 _! 

force, or the magnetic flux, has the dimension MJ L^ T 
hence the unit difference of electromagnetic potential, 
called also potential and electromotive force, has the dimen- 
sion W* L* T 2 . 

B. Ratio of the Units. — It can be observed from Table 
A that the dimensions of similar units in the electrostatic 
and the electromagnetic systems are different. Thus the 

1 3 -l 

electrostatic unit of quantity has the dimension M^ L 2 " T 

while the electromagnetic unit has the dimension M^ L" 3 " 
This difference, which exists throughout the two series 
arises from faulty definitions of the electrostatic unit of 



APPENDIX. 



149 



quantity in the one case and the magnetic unit of quantity 
(strength of pole) in the other. Every electrostatic field 
is set up in some medium that has a dielectric constant h 
(called also specific inductive capacity) and every mag- 
netic field is set up in some medium that has a permeability 
(i. Neither the dimension nor the absolute value of either 
of these factors is known, and the values of h and tx in air 
are taken as unity in lieu of better units. If the dimensions 



TABLE B. 







Dimensions. 




Ratio op 
Units. 


Name of 
Unit. 


Electrostatic. 


Electromagnetic. 


Electrostatic. 
Electromagnetic. 


c.g.s. static 


Pract. mag- 
netic. 


Quantity. . 
Potential . . 


13-1 
]VP" hi T~ l 


1 1 
M2LI 
1 3 _„ 

M3 L2 T 2 


LT 1 = v 
I/ 1 T = v_1 


i X 10-9 
300 


Current . . . 


1 3 _o 

]VP" L^ T A 


M^ li T 1 


L T" 1 = v 


i x 10- 9 


Resistance . 


L _1 T 


LT 1 


t-2 rr.2 _ V ~2 


9 X 10 U 


Capacity. . . 


L 


L l T 2 


l 2 r 3 = v 2 


1 X lO" 11 



Note.— The last entry is for farads. For microfarads, 



c. g. 



static , 



microfarads 900,000 



and absolute value of these factors were known, and they 
were properly introduced into the definitions of their re- 
spective systems, the dimensions and values of the corre- 
sponding terms of the two sytems would be in accord. 
Furthermore it is irrational that the dimension of a 

1 3 

unit should contain such terms as M 2 or L f since these 
terms have no physical meaning. Rucker has shown that 
the dimensions of the units could be rationalized and the 



150 STATIC ELECTRICITY. 

two systems brought into accord if h and ^ were properly 
introduced into the definitions and if their product were 
assumed to have the dimension of the reciprocal of the 
square of a velocity. 

In Table B are given the ratios of the dimensions of 
similar units in the two systems. It may be noted that in 
these ratios only the terms L and T appear and that these 
invariably appear with equal and opposite indices. That 
is, the ratio in every case is some power of the dimension 
of a velocity, which may be represented by v. This is a 
definite concrete velocity and its value is this : two par- 
allel wires carrying currents in the same direction are 
mutually attracted. An electrostatic charge being moved 
through space produces to a certain extent the effect of a 
current of electricity flowing along the path of its motion. 
The extent of this effect depends upon the velocity with 
which the charge is moved. Two similar electrostatic 
charges are mutually repelled. The velocity v which enters 
into the ratios of the dimensions of the two systems is that 
velocity at which two quantities of electricity should travel 
side by side so that their mutual electromagnetic attraction 
will be just equal to their mutual electrostatic repulsion. 

The numerical value of v in centimeters per second has 
been experimentally determined by several investigators. 

Weber and Kohlrausch compared the two units of quan- 
tity and gave 

v = 3.1074 xlO 10 . 

Kelvin compared the two units of potential and found 
v = 2.825 x 10 10 . 

Clerk Maxwell balanced an electrostatic attraction 
against an electrodynamic repulsion and gave 
v = 2.88 x 10 10 . 



APPENDIX. 151 

Ayrton and Perry, by measuring the capacity of an air 
condenser eleetrodynamically and determining its electro- 
static capacity from consideration of its dimensions found 
<v = 2.980 x 10 10 . 

These values all agree very closely with the velocity of 
light in vacuo, which is approximately 3 x 10 10 centimeters 
per second. This is a most important fact in support of 
the electromagnetic theory of light. 

In Table B are also given the ratios of the c. g. s. electro- 
static units to the practical electromagnetic units, assuming 
v = 3 x 10 10 . In deriving these ratios it will be noticed 
that the ratio of the absolute units is the inverse ratio of 
their dimensions. This at first glance may seem para- 
doxical, but the following explanation of the case of the 
units of quantity may serve to make this matter clear. 

.1 3. _i 

The dimension of the static unit of quantity is M~ a ^ 2 T , 

that of the dynamic unit JVL 2 L 2 . It is therefore evident 
that some system of units of length, mass, and time can be 
chosen so that the actual value of the two units of quantity 
will be the same notwithstanding the difference in the 
naming of their dimensions. In this new system then let 
the 

unit of length equal L centimeters, the 
unit of mass equal M grams, and the 
unit of time equal T seconds. 
The static unit of quantity in this new system will contain 

1 3 -l 
M^L^T of the c. g. s. static units of quantity, and like- 
wise the dynamic unit will contain M^ L 2 of the c. g. s. 
dynamic units. But by our preliminary assumption these 



152 STATIC ELECTRICITY. 

two unit quantities in the new system are equal. There- 
fore 

W> U T 1 X one c. g. s. static unit = MJ U X one 



c. g. s. dynamic unit. 



or 



one c. a. s. static unit M^~ L" 2 " 



L _1 T 



one c. g. s. dynamic unit ]yjr jj 'j rl 
Now v is a definite concrete velocity, entirely independent 
of the size of the fundamental units. In the new system 
which we assumed above v is evidently that velocity which 
would be designated by unity. In the c. g. s. system v is 
found by experiment to be 3 X 10 10 . In any system whatso- 
ever the ratio 

electrostatic unit of quantity _ 1 
electrodynamic unit of quantity v 

holds true. This, it is now observed, is the reciprocal of 
the ratio of the dimensions of the quantities. 

In a like manner it can be shown for each of the other 
pairs of units- in the two systems, that the ratio of the units 
is the inverse ratio of their dimensions. 

The above reasoning is adapted from Professor J. D. 
Everett's " Units and Physical Constants," to which work 
the student is referred for further information as to the 
values and relations of the c. g. s. units. 



INDEX. 



[The figures refer to page numbers.] 



Absolute condensers, 65. 

electrometers, 93. 

units, 143. 
Air condensers, 65. 
Amalgam, Kienmayer's, 120. 
Appropriating brushes, 122. 
Attraction, 1. 

due to electrified plane, 44. 
Attracted disc electrometer, 93. 
Ayreton and Mather's electrom- 
eter, 103. 

Ballistic galvanometer, 84. 
Biot's spheres, 36. 
Bound and free charges, 14. 
Brush discharge, 141. 
Brushes, appropriating, 122. 
neutralizing, 133. 

Calculation of capacity, 58. 
Calibration of galvanometer, 91. 
Capacity. 46. 
calculation of, 58. 
comparison of, 77. 
measurement of, 73. 
specific inductive, 52. 
of coaxal cylinders, 61. 
concentric spheres, 59. 
condenser, formula for, 72. 
earth, 47. 
Ley den jar, 51. 
lines, 63. 

parallel plates, 59. 
plate condenser, 51. 
sphere, 46. 
Capillary electrometer, 104. 
Carhart's electrometer, 102. 
Carriers. 121. 
Charge, definition of, 3. 
energy of, in condenser, 64. 
in a cavity, 37. 
residual, 54. 
within a conductor, no, 35. 



153 



Charges always equal and oppo- 
site, 6. 
Cohn and Arons, method of, 78. 
Collecting combs, 122, 133. 
Combs, discharging or collecting, 

122, 133. 
Comparison of capacities, 77. 
Condenser, 47. 

dielectrics, 57. 

energy in, 64. 

principle of, 47. 

seat of charge in, 52. 
Condensers, commercial, 65. 

formula for capacities of, 72. 

in parallel, 70. 
series, 72. 
Condensing electroscope, 49. 
Conductive series, 9. 
Conductors and insulators, 8. 
Connection of condensers, 70. 
Constant of galvanometer, 91, 93. 
Coulomb's law, 20. 
Current inductor method, 91. 

Damping, 85. 

Damped galvanometer, equation 

for, 86. 
D'Arsonval galvanometer, 84. 
Derived units, 143. 
De Sauty's method, 79. 
Dielectric coefficient, 53. 

hysteresis, 56. 

stresses, 54. 
Discharges from static machines, 

137. 
Discharging combs, 122, 133. 
Difference of potential, 29. 

measure of, 45. 

Earth as a condenser, 47. 

inductor method, 91. 
Ebonite as an insulator, 10. 
Elastic limit of dielectrics, 55. 



154 



INDEX. 



[The figures refer to page numbers.] 



Electric expansion, 55. 

fly. 25. 

wind, 24. 
Electrification, definition of, 3. 

vitreous and resinous, 6. 

positive and negative, 6. 
Electrodes, 122. 
Electrodynamometer, 84. 
Electrolytic condenser, 68. 

electrometer, 104. 
Electrometer, absolute, 93. 

Ayreton and Mather's, 103. 

capillary, 104. 

Carhart's, 102. 

Ryan's, 100. 

quadrant. 96. 
Electrophorous, 15. 
Electroscope, 17. 

condensing. 49. 
Electrostatic field, 26. 

ground detectors, 112. 
Stanley, 112. 
General Electric, 113. 

series, 7. 

voltmeters, 109. 
Kelvin's, 109. 
multicellular, 110. 
Stanley, 110. 
Thomson's, 112. 
Energy in charge, 3. 

in condenser, 64. 
Equation for damped galvanome- 
ter, 86. 
Equipotential surface, 29 

line, 29. 

Faraday's bag, 37. 

ice pail experiment, 38. 
Field, electrostatic, 26. 

intensity of, 27. 

plates, 122. 
Fiy, electric, 25. 
Free and bound charges, 14. 
Friction machines, 117. 
Fundamental units, 143. 

Galvanometer, ballistic, 84. 
calibration of, 92. 
constant, 91, 93. 



General Electric ground detector, 

113. 
Generators of high potential: 

discharges from, 137. 

dropping, 136. 

friction, 117. 

Holtz, 126. 

influence, 120. 

Toepler, 1*0. 

Wimshurst. 132. 
Glass an insulator, 11. 
Gold leaf electroscope, 17. 
Ground detectors: 

Stanley, 112. 

General Electric, 113. 

Hard rubber as an insulator, 10. 
Hetrostatic method for electrom- 
eters, 96, 98. 
High potential condensers, 68. 
Holtz machine, 126. 
Hysteresis in dielectrics, 56. 

Ice pail experiment, 38. 
Idiostatic method for electrom* 

eters, 95, 97. 
Induction (influence), 14. 
Inductivity, 53. 
Inductors (field plates), 122. 
Influence, definition of, 12. 
Insulators and conductors, 8. 
Intensity of field, 27. 

near charged surface, 41. 

Kelvin's dropping generator, 136. 

electrometer, 93. 

electrostatic voltmeter, 109. 

quadrant electrometer, 96. 
Kerosene as an insulator, 11. 
Kienmayer's amalgam, 120. 

Ley den jar, 49. 

seat of charge in, 52. 
Lines of force, 26. 

Mica as an insulator, 11. 

condensers, 65. 
Measurement of capacity, 73. 



INDEX. 



155 



[The figures refer to page numbers.] 



Motors, electrostatic, 184. 
Multicellular voltmeters, 110 

Neutralizing brush, 133. 
rod, 122. 

Oils as insulators, 11. 
One fluid theory, 3. 
Ozokerite as an insulator, 11. 

Paper condensers, 67. 
Paraffin as an insulator, 11. 
Petroleum as an insulator, 11. 
Positive and negative charges, 5. 
Potential, electrostatic, 28. 

definition of, 29. 

unit of, 32. 
Proof plane, 36. 

Quantity, definition of, 21. 

of charge, 3. 

units of, 21, 145. 
Quadrant electrometers, 96. 

Oarhart's, 102. 

Kelvin's, 97. 

Ryan's, 100. 
Quartz as an insulator, 11. 

Ratio of units, 148. 
Residual charge, 54. 
Resinous electrification, 6. 
Repulsion, 4. 
Ryan's electrometer, 100. 

Seat of charge in condenser, 52. 
Separation of charge, method 
of, 82. 



Series, conductive, 9. 

electrostatic, 7. 

Volta's (contact), 8. 
Shellac as an insulator, 12. 
Specific inductive capacity, 52. 

table of, 53. 
Standard condensers, 65. 

calibration of galvanometer 
by, 92. 
Stanley static ground detector, 112. 

voltmeter, 1 10. 
Stress in dielectric, 54. 

on electrified surface, 42. 
Surface density, definition of, 22. 

tension of fluids altered, 4. 

Thomson's method. 80. 
static voltmeter, 112. 
Toepler machine, 120. 
Torsion balance, 20. 
Tubes of force, 39. 
Tuning fork contact maker, 74. 
Two fluid theory, 13. 

Units, definitions of, 143. 
dimensions of, 143. 
ratios of, 148. 

Vitreous electrification, 6. 

Volta's series, 8. 

Voltmeters, electrostatic, 1C9. 

multicellular, 110. 

Stanley, 110. 

Thomson's, 112. 
Vulcanite as an insulator, 10. 

Wimshurst machine, 132. 
Wind, electric, 24. 



My 9 »(H 



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